L.   P.   SHIDY 


AN 


INTRODUCTION 


TO 


NATURAL  PHILOSOPHY; 


DESIGNED  AS  A 


TEXT-B  OOK 


FOB  THE   USE   OF 


STUDENTS    IN    COLLEGE 


BY  DENISON  OLMSTED,  LL.D., 

LATE    PROFESSOR    OP    NATURAL    PHILOSOPHY    AND    ASTRONOMY    IN    YALE    COLLEGE. 


SECOND  REVISED  EDITION 

BY  E.    S.    SNELL,    LL.D., 

PROFESSOR  OF  MATHEMATICS   AND   NATURAL  PHILOSOPHY  IN  AMHERST  COLLEGE.. 


NEW  YORK : 

C  O  I,  L I  N  S     &     BROTHER, 

106  LEONARD  STREET. 

1870. 


Entered  according  to  Act  of  Congress,  in  the  year  1844,  by 

DENISON  OLMSTED, 
In  the  Clerk's  Office  of  the  District  Court  of  Connecticut. 


REVISED  EDITION. 
Entered  according  to  Act  of  Congress,  in  the  year  1860,  by 

JULIA  M.  OLMSTED, 

FOR  THE  CHILDREN  OF  DENISON  OLMSTED,  DECEASED, 
In  the  Clerk's  Office  of  the  District  Court  of  the  District  of  Connecticut. 


SECOND  REVISED  EDITION. 
Entered  according  to  Act  of  Congress,  in  the  year  1870,  by 

JULIA  M.  OLMSTED, 

FOR  THE  CHILDREN  OF  DENISON  OLMSTED,  DECEASED, 
In  the  Office  of  the  Librarian  of  Congress,  at  Washington. 


IN  MEMORfAAf 

L.V.  SW.A 


Electrotyped  by  SMITH  &  MCDOUGAL.  82  Beebman  Street. 


PREFACE. 


THE  object  kept  in  view  in  the  present  revision  is  the  same 
as  heretofore — to  prepare  a  book  suitable  for  use  in  the 
College  recitation-  room.  The  work  does  not  aim  to  be  a  hand- 
book of  Physics,  giving  information  on  all  points  relating  to  the 
subjects  treated  of,  but  merely  a  book  of  first  principles,  accom- 
panied by  a  sufficient  number  of  illustrative  statements  and  dia- 
grams to  render  those  principles  clear,  and  to  impress  them  on 
the  memory. 

The  more  prominent  departures  from  the  former  revision  are 
the  following: 

1.  Most  of  the  second  part  of  Mechanics  is  omitted ;  and  what 
is  retained  is  introduced  in  appropriate  connections  in  Part  L 
The  second  part  was  originally  intended  as  a  sort  of  substitute 
for  a  course  of  experimental  lectures.    But  colleges  are  now  so 
generally  supplied  with  apparatus  for  illustration,  that  it  seems 
unnecessary  to  encumber  the  volume  with  information  which  can 
be  presented  so  much  more  satisfactorily  by  the  lecturer. 

2.  Instead  of  the  brief  Part  entitled  "Electro-magnetism," 
first  presented  in  the  work  when  the  former  revision  was  made, 
the  subject  of  "  Dynamical  Electricity"  is  now  discussed  as  fully 
as  the  other  branches. 

3.  The  subject  of  "Heat"  for  the  first  time  forms  one  Part  of 
the  work,  although  the  course  of  instruction  in  many  colleges 
still  retains  it  in  the  chemical  department. 

4.  Some  additions  are  made  to  the  applications  of  the  differ- 
ential and  integral  calculus,  and  all  the  discussions  of  this  char- 
acter are  brought  together  in  an  "Appendix"  at  the  close  of  the 
volume. 

5.  More  than  three-fourths  of  the  engravings  are  new,  most 
of  which  were  drawn  expressly  for  this  revision. 


iy  PREFACE. 

Besides  these  more  apparent  alterations,  it  should  be  added 
that  a  large  part  of  the  whole  book  has  been  carefully  rewritten, 
and  additions  and  improvements  made  in  almost  every  page  of  it. 

The  author  of  the  revision  wishes  to  express  his  indebtedness 
to  Professor  JOSEPH  FICKLIK,  of  the  University  of  Missouri,  who 
has  furnished  much  valuable  material  for  the  Part  on  "  Mechan- 
ics," and  has  critically  examined  the  mathematical  portions  of  the 
book,  and  rendered  essential  aid  in  correcting  and  improving 
them.  The  Part  on  "  Dynamical  Electricity  "  is  almost  entirely 
the  work  of  Professor  CHARLES  H.  SMITH,  of  Cincinnati,  Ohio, 
formerly  Tutor  of  Natural  Philosophy  in  Yale  College.  Much 
credit  is  due  to  him  for  presenting  the  principles  of  that  exten- 
sive and  complex  branch  of  physics  in  a  clear  and  systematic 
form.  Dr.  B.  JOT  JEFFRIES,  of  Boston,  Lecturer  on  Optical  Phe- 
nomena and  the  Eye  in  Harvard  University,  has  kindly  assisted 
in  preparing  the  description  of  the  eye  and  its  adjustments,  and 
has  allowed  his  own  original  drawings  to  be  copied  in  the  engrav- 
ings relating  to  this  subject. 

E.  S.  SSTELL. 

AMHERST  COLLEGE,  September,  1870. 


CONTENTS. 


INTRODUCTION. 

PAGE 

Classification  of  the  Physical  Sciences.— Definitions  relating  to  matter.— 
Properties  of  matter.— Branches  of  Natural  Philosophy 1—3 


PART  I.— MECHANICS. 

CHAPTER  I. 

Motions  classified. — Uniform  motion. — Momentum. — Forces  classified. — 
Laws  of  motion. — Gravity. — Its  laws. — Questions 4 — 12 

CHAPTER  II. 

Uniform  and  variable  motion  represented  geometrically. — Laws  of  the 
fall  of  a  body. — Motion  of  a  body  projected  up  or  down. — Formulae  for 
the  fall  of  bodies. — Space  and  time  represented  by  co-ordinates. — At- 
wood's  machine. — Living  force. — Questions 12 — 23 

CHAPTER  III. 

Two  or  more  forces  acting  on  a  body. — Parallelogram  of  forces. — Triangle 
of  forces. — Polygon  of  forces. — Curvilinear  motion.— Calculation  of  the 
resultant. — Examples.— Resolution  of  motion. — Resultant  found  by 
rectangular  axes. — Analytical  expression  for  the  resultant — Principle 
of  moments. — Parallel  forces. — Parallelepiped  of  forces. — Rectangular 
axes. — Equilibrium  of  forces. — Couples. — Forces  resisted  by  a  smooth 
plane ....  23—42 

CHAPTER  IV. 

Centre  of  gravity  defined. — Centre  of  two  equal  bodies. — Of  two  unequal 
bodies. — Equal  moments. — Three  or  more  bodies. — Triangle. — Poly- 
gon.— Perimeters. — Pyramid. — Examples — Centre  of  gravity  referred 
to  a  point  or  plane. — Trapezoid. — Centrobaric  mensuration. — Stable, 
unstable,  and  neutral  equilibrium. — Motion  of  the  centre  of  gravity  of 
a  system. — When  one  body  moves,  and  when  more  than  one  move. — 
Examples 42—58 

CHAPTER  V. 

Collision  of  inelastic  bodies. — Formulae. — Questions. — Collision  of  elastic 
bodies,  equal,  unequal. — Series  of  bodies,  equal,  decreasing,  increas- 
ing.— Living  force  lost  in  the  collision  of  inelastic  forces. — Preserved  in 
the  collision  of  elastic  bodies. — Impact  on  a  plane 58 — 65 

CHAPTER  VI. 

Machines  classified.— Three  orders  of  straight  lever.— Equal  moments.— 
Compound  lever. — Balance. — Steelyard. — Platform  scales. — Wheel  and 
axle. — The  same  compounded. — Connection  by  teeth  and  by  bands. — 
Pulley. —  Fixed,  movable,  compound. —  Rope  machine.  —  Branching 
rope. — Funicular  polygon. — Inclined  plane. — Relation  of  power,  weight, 
and  pressure. — Body  between  two  planes. — Equilibrium  of  bodies  on 
two  planes. — Screw. — Combined  with  the  lever. — Endless  screw — Right 
and  left  hand  screw. — Wedge. — Knee-joint. — Principle  of  virtual  veloci- 
ties.— Friction  in  machinery. — Laws  of  sliding  friction. — Friction  of 
axes. — Rolling  friction. — Friction  wheels 65 — 95 


vi  CONTENTS. 

CHAPTEE  VII. 

The  fraction  of  gravity  on  an  inclined  plane. — Formulae  for  descent. 

Descent  on  the  chords  of  a  circle. — Series  of  planes.- -Descent,  on  a 
curve. — The  pendulum. — Calculation  of  its  length. — Point  of  suspen- 
sion and  centre  of  oscillation  interchangeable. — The  cycloid. — Its  prop- 
erties.— Descent  on  a  cycloid. — The  involute  of  a  semi-cycloid. — The 
cycloidal  pendulum. — Results  applied  to  common  pendulum. — Compen- 
sation pendulum 95 — 108 

CHAPTER  VIII. 

Formulae  for  the  path  of  a  projectile. — Different  angles  of  elevation  for 
the  same  range. — The  greatest  height  of  a  projectile  — Formulas  for  a 
horizontal  plane. — Equation  of  the  path  of  a  projectile. — Range  on  an 
oblique  plane. — Questions' — Central  forces  described — Expressions  for 
centrifugal  force  in  circular  motion. — Two  bodies  revolving  about  their 
centre  of  gravity. — Centrifugal  force  on  the  earth. — Examples. — Com- 
position of  two  rotary  motions — The  gyroscope. . .  109 — 120 

CHAPTEE  IX. 

Longitudinal  strength. — Lateral  strength,  both  ends  supported. — Stress 
from  weight  of  the  beam. — Weight  at  the  centre  of  the  beam. — 
Weight  at  any  other  point. — Form  for  equal  strength. — Lateral 
strength,  one  end  supported. — Prismatic  beams  breaking  by  their  own 
weight. — Structures  weaker  as  they  are  larger. — Solid  and  hollow  cyl- 
inders.— Miscellaneous  problems  in  Mechanics 120—132 


PAKT  II— HYDROSTATICS. 

CHAPTER  I. 

Liquids  distinguished  from  solids  and  gases. — Transmitted  pressure. — 
Hydraulic  press. — Equilibrium  of  a  liquid. — Curvature  of  surface. — 
Spirit  level  — Pressure  as  depth. — Amount  of  pressure. — Artesian 
wells. — Centre  of  pressure. — Loss  of  weight  in  water. — Equilibrium 
of  floating  bodies  — Specific  gravity  — Hydrometer. — Magnitude  found 
by  specific  gravity.  —  Cohesion  and  adhesion.  —  Capillary  action. — 
Liquids  raised  and  depressed  by  the  side  of  a  solid. — Capillary  tubes 
and  plates. — Effect  of  capillarity  on  floating  bodies 133 — 152 

CHAPTEE  II. 

Depth  and  velocity  of  discharge. — Descent  of  surface. — Orifices  in  differ- 
ent situations. — Vena  contracta. — Friction  in  pipes. — Jets. — Rivers. — 
Lifting  pump. — Chain  pump.— Hydraulic  ram. — Water-wheels. — Tur- 
bine.— Barker's  mill. — Resistance  of  a  liquid. — Waves  of  oscillation. — 
Molecular  movements.— Sea  waves.— Waves  of  translation 153—160 


PART   III.— PNEUMATICS. 

CHAPTEE  I. 

Nature  of  gases. — Mariotte's  law. — The  air-pump. — Rate  of  exhaustion. — 
Air  condenser. — Torricelli's  experiment. — Atmospheric  pressure. — Ba- 
rometer.-r-Pressure  at  different  latitudes. — Diurnal  variation. — Effect 
of  weather  on  pressure. — Heights  measured  by  the  barometer. — Gauges 
of  the  air-pump „ 167 — 176 

CHAPTEE  II. 

Bellows.  —  Siphon,  —  Suction  pump.  —  Forcing-pump.  —  Fire-engine  — 
Hero's  fountain. — Manometer. — Apparatus  for  preserving  a  water- 
level 177—182 


CONTENTS.  Vii 

CHAPTER  III. 

Quantity  of  the  atmosphere. — Its  height. — Change  of  density  with 
height. — Trade  winds. — Return  currents. — Wind  of  higher  latitudes. — 
Land  and  sea  breezes. — Current  in  a  medium. — Effect  on  a  surface. — 
Vortices 182—188 


PART   IV.— SOUND. 

CHAPTER  I. 

Vibrations  the  cause  of  sound. — Sonorous  bodies. — Air  the  common  me- 
dium.— Velocity  in  air. — Diffusion  of  sound. — Nature  of  the  waves. — 
Gases,  liquids,  and  solids  as  media  of  sound. — Mixed  media 189 — 196 

CHAPTER  II. 

Laws  of  reflection  of  sound. — Echoes. — Concentrated  echoes. — Resonance 
of  rooms  — Halls  for  public  speaking. — Refraction  of  sound. — Inflection 
of  sound 197—202 

CHAPTER  III. 

Vibrations  in  musical  sounds. — Pitch. — The  monochord. — Formula  for 
time  of  a  vibration — The  number  of  vibrations  in  a  given  time. — Vi- 
brations of  a  column  of  air. — Modes  of  vibration  in  different  pipes. — 
Vibration  in  parts. — Modes  of  exciting  the  vibrations.— Rods  and 
laminge. — Chladni's  plates. — Bells. — The  voice. — The  organ  of  hear- 
ing   202—214 

CHAPTER  IY. 

Numerical  relations  of  musical  sounds. — Repetition  of  the  scale. — Modes 
of  naming  the  notes. — Diatonic  and  chromatic  scales. — Chords  and  dis- 
cords,— Temperament. — Harmonics. — Overtones. — Effect  on  quality  of 
tone .  — Communication  of  vibrations .  — Crispations .  — Interference .  — 
Number  and  length  of  waves  for  each  note. — Vibrations  visibly  pro- 
jected 214-223 


PART  V.— MAGNETISM. 

CHAPTER  I. 

Natural  and  artificial  magnet. — Attraction  of  iron. — Polarity.— Mutual 
action  of  magnets. — Magnetic  induction. — Reflex  influence. — Double 
induction. — Coercive  force. — Magnetism  not  transferred. — Law  of 
force  and  distance. — Positions  of  a  needle  near  a  magnet.— Magnetic 
curves •  •  •  •  224 — 231 

CHAPTER  II. 

Declination  of  the  needle. — Isogonic  curves. — Variations,  secular,  annual, 
and  diurnal. — Dip  of  the  needle. — Isoclinic  curves. — Magnetic  inten- 
sity.— Isodynamic  curves. — Magnetic  observatories. — Aurora  Borealis. — 
Source  of  the  earth's  magnetism. — Formation  of  permanent  magnets. — 
The  declination  compass. — The  mariner's  compass. — The  needle  ren- 
dered astatic. — Theory  of  magnetism 231 — 241 


PART  VI.— FRICTIONAL  OR  STATICAL  ELECTRICITY. 
CHAPTER  I. 

Definitions. — Electroscopes.— Two  electrical  states .  — Theories.-*-Mutual 
action. — Conduction. — Modes  of  insulating — Sphere  of  communication 
and  of  influence 242—247 


viii  CONTENTS. 

CHAPTER  II. 

Plate  machine. — Cylinder  machine. — Hydro-electric  machine. — Phenom- 
ena of  the  machine. — Torsion  balance. — Law  of  force. — Charge  on  the 
surface. — How  distributed. — Held  by  the  air 247 — 253 

CHAPTER  III. 

Elementary  experiment  on  induction.— Reaction. — Effect  of  dividing  the 
conductor. — Of  lengthening  it. — Disguised  electricity. — Series  of  con- 
ductors.— Why  an  unelectrified  body  is  always  attracted. — The  Franklin 
plate. — The  Leyden  jar. — Its  theory. — Spontaneous  discharge. — Series 
of  jars. — Dividing  a  charge. — Use  of  coatings. — The  free  part  of  the 
charge  explained.-— Vibrations  and  revolutions. — Residuary  discharge. — 
Battery.— Discharging  electrometers. — Why  a  point  held  toward  a 
charged  body  discharges  it. — Induction  applied  to  the  explanation  of 
electroscopes,  condenser, and  electrophorus. — Induction  machine..  253 — 266 

CHAPTER  IV. 

Effects  of  discharges. — Luminous  effects. — Colors. — Luminous  figures. — 
Mechanical,  chemical,  and  physiological  effects. — Velocity  of  Elec- 
tricity   266—270 

CHAPTER  V. 

Electricity  in  the  air. — Thunder  storms. — Lightning  is  a  discharge  of 
electricity. — Rods. — How  they  protect. — Protection  of  the  person. — 
How  lightning  causes  damage 271—275 


PART  VII.— DYNAMICAL  ELECTRICITY. 
CHAPTER  I. 

Electricity  by  chemical  action. — An  element. — A  battery. — Comparison 
of  statical  and  dynamical  electricity. — Quantity  batteries,  intensity  bat- 
teries   27  6—280 

CHAPTER  II. 

Helices. — The  solenoid. — Ampere's  theory — Mutual  action  of  currents. — 
Relations  of  currents  and  magnets. — The  galvanometer. — The  earth's 
polarity. — Thermo-electricity.— Magnetic  induction  by  currents. — The 
U-magnet. — Its  power  to  sustain  a  weight 280 — 288 

CHAPTER  III. 

Currents  inducing  currents. — Characteristics  of  induced  currents. — Both 
currents  in  one  wire. — Names  of  these  circuits  and  currents. — Coils, 
primary  and  secondary. — Ruhmkorff's  coil. — One  coil  moved  into  and 
out  of  another. — Magneto-electricity.— ^Explained  on  Ampere's  theory. — 
Clarke's  magneto-electric  machine. — Its  operation 289 — 298 

CHAPTER  IV. 

Practical  applications.— Electrolysis .  —  Electro-plating.— Electrotyping.— 
Electric  light  and  heat. — Mechanical  movements. — Electro-magnetic 
machine.— Electro-magnetic  telegraph. — Its  parts. — Its  operation. — Re- 
peaters.— Ocean  cable. — Fire-alarm. — Chronograph 298 — 309 


.  PART  VIII.— HEAT. 

CHAPTER  I. 

Nature  of  h^at.— Expansion  by  heat,  contraction  by  its  loss. — Thermom- 
eter.— Different  scales. — Pyrometer. — Coefficient  of  expansion. — Ther- 
mal force  very  great.— Case  of  contraction  by  heat 310 — 315 


CONTENTS.  jfc 

CHAPTER  II. 

Heat  communicated  in  several  ways. — Radiation. — Heat  tends  to  an  equi- 
librium.— Reflection. — Concentration  by  reflection. — Absorption. — Con- 
duction.— Effect  of  molecular  arrangement  in  solids. — Convection. — 
Diathermancy 315—820 

CHAPTER  III. 

Specific  heat. — Method  of  finding  it.— Change  of  condition. — Latent 
heat. — Boiling  under  pressure. — Freezing  by  melting. — Spheroidal  con- 
dition  321—325 

CHAPTER  IV. 

Force  of  steam. — Change  of  tension  with  temperature. — Steam-engines. — 
The  engine  of  Watt. — Single  and  double  acting. —  Low-pressure  en- 
gine.— Steam-valves. —  High-pressure  engine. — Applications  of  steam- 
power. — Mechanical  equivalent  of  heat  325 — 331 

CHAPTER   V. 

How  the  air  is  warmed. — Limit  of  perpetual  frost. — Isothermal  lines. — 
Moisture  of  the  air. — Dew  point. — Measure  of  vapor. — Hygrometers. — 
Forms  of  precipitation  of  vapor. — Dew,  frost. — Fog,  cloud. — Classifica- 
tion of  clouds. — Rain,  mist,  hail,  sleet,  snow. — Theories  of  precipita- 
tion.— Cyclones. — Draft  of  flues. — Ventilation  of  rooms,  of  mines. — 
Sources  of  heat 331—341 


PART    IX.— LIGHT. 

CHAPTER  I. 

Light  moves  in  straight  lines. — Its  velocity. — Intensity  at  different  dis- 
tances.—Loss  by  absorption. — Photometers. — Shadows 342 — 346 

CHAPTER  II. 

Reflection. — Its  law. — Inclination  of  rays  not  changed  by  a  plane  mir- 
ror.— Rays  converged  by  a  concave,  diverged  by  a  convex  mirror. — 
Conjugate  foci. — Images  by  a  plane  mirror. — Object  and  image  symmet- 
rical.— Space  on  the  mirror  occupied  by  the  image. — Displacement  by 
two  reflections. — Multiplied  images  by  two  mirrors,  parallel,  inclined. — 
Images  by  a  concave  mirror,  by  a  convex  mirror. — Caustics  by  reflec- 
tion.—Spherical  aberration  of  mirrors 346—360 

CHAPTER  III. 

Refraction. — Law  as  to  density. — Law  as  to  inclination. — Limit  of  emer- 
gence from  a  denser  medium. — Transmission  through  plane  surfaces, 
parallel,  inclined. — Multiplying  glass. — Refractive  power  found  by  a 
prism. — Light  through  one  surface,  plane,  convex,  concave. — Lenses. — 
Effect  of  the  convex  lens,  of  the  concave  lens. — Optic  centre. — Conju- 
gate foci. — Images  by  the  convex  lens,  by  the  concave  lens. — Caustics 
by  refraction. — Spherical  aberration  of  lenses. — Remedy.— Atmospheric 
refraction.— Mirage 360—374 

CHAPTER  IV. 

The  prismatic  spectrum. — Colors  recombined. — Complementary  colors. — 
Fraunhofer  lines. — Lines  of  terrestrial  substances  burning. — Theory  of 
lines  in  the  solar  spectrum. — Dispersion  of  light. — Chromatic  aberra- 
tion of  lenses.1— Achromatism 375 — 381 

CHAPTER  V. 

The  rainbow. — Experiment  with  a  sphere  of  water. — Course  of  rays  in 
the  primary  bow. — In  the  secondary  bow. — Axis  of  the  bows. — Their 
circular  form. — Colors  of  the  two  bows  in  contrary  order. — The  tertiary 
bow  — The  common  halo. — Caused  by  crystals  of  ice. — Its  frequency. — 
The  mock-sun 381—388 


X  CONTENTS. 

CHAPTER  VI. 

Natural  colors  of  bodies. — Inflection  of  light. — Breadth  of  fringe  varies 
with  the  color. — Why  not  always  seen  on  the  edges  of  bodies. — Color 
by  striation. — By  thin  laminae. — Ratio  of  thicknesses  for  the  successive 
rings. — Mode  of  finding  the  thickness. — Newton's  rings  by  a  mono- 
chromatic lamp 388 — 393 

CHAPTER  VII. 

Double  refraction. — Iceland  spar. — Ordinary  and  extraordinary  ray. — 
Optical  relations  of  the  axis. — Crystals  of  positive  and  of  negative 
axis. — Polarization  of  light. — By  reflection. — Polarizing  and  analyzing 
plates. — By  bundle  of  plates. — By  absorption. — By  double  refraction. — 
Every  polarizer  an  analyzer. — Color  by  polarized  light 393 — 399 

CHAPTER  VIII. 

The  wave  theory. — Its  postulates. — Reflection  according  to  it. — Refrac- 
tion according  to  each  theory. — Interference. — By  thin  plates. — Bj  two 
mirrors. — By  inflection. — Length  of  waves  and  number  per  second  for 
each  color. — Modp  of  vibration  in  polarized  light. — Application  in  the 
several  modes  of  polarizing 399 — 407 

CHAPTER  IX. 

Image  by  light  through  an  aperture. — Effect  of  a  convex  lens  at  the  aper- 
ture.— The  eye. — Parts  of  the  interior. — Vision. — Adaptations. — Ac- 
commodation.— How  caused. — Long-sightedness. — Short-sightedness. — 
Cause  of  each. — Why  an  object  is  seen  erect  and  single. — ^ndirect 
vision. — The  blind  point, — Continuance  of  impressions. — Accidental 
colors. — Estimate  of  distance  and  size  by  the  eye. — Binocular  vision. — 
The  stereoscope 407—415 

CHAPTER  X. 

The  camera  lucida. — The  microscope. — The  single  microscope. — Limit  of 
its  power. — The  compound  microscope. — Its  magnifying  power. — Im- 
provements in  its  construction. — Microscopes  for  projecting  images. — 
The  magic  lantern. — The  solar  microscope. — The  astronomical  tele- 
scope.— Its  powers. — Mode  of  mounting. — The  terrestrial  telescope. — 
Galileo's  telescope. — The  Gregorian  telescope. — The  Herschelian  tele- 
scope  416—424 


APPENDIX. 

APPLICATIONS  OF  THE  CALCULUS. 
I.  FALL  OF  BODIES. 

Differential  equations. — Fall  through  small  distances  near  the  earth. — 
Through  great  distances. — Method  of  finding  velocity  and  time. — Fall 
within  the  earth.— Velocity  and  time  found 425 — 428 

II.  CENTRE  OF  GRAVITY. 

Principle  of  moments. — Formula  prepared. — Applications  of  formulae  to 
various  cases 428 — 432 

III.  CENTRE  OF  OSCILLATION. 

Moment  of  inertia  for  any  axis. — Examples. 432 — 434 

IV.  CENTRE  OF  HYDROSTATIC  PRESSURE. 

General  formula?.— Examples 434—436 

V.  ANGULAR  RADIUS  OF  THE  RAINBOW  AND  THE  HALO. 

The  primary  bow. — The  secondary  bow. — The  halo 436 — 437 


NATURAL    PHILOSOPHY. 


INTRODUCTION. 

Art.  1.  Classification  of  Physical  Sciences. — The  ma- 
terial world  consists  of  two  parts — the  organized,  including  the 
animal  and  vegetable  kingdoms;  and  the  unorganized,  which 
comprehends  the  remainder.  Organized  matter  is  treated  of  in 
Physiology,  and  in  those  branches  of  science  usually  called  Natural 
History.  Unorganized  matter  forms  the  subject  of  Natural  Phi- 
losophy and  Chemistry.  Chemistry  considers  the  internal  consti- 
tution of  bodies,  and  the  relations  of  their  smallest  parts  to  each 
other.  Natural  Philosophy  deals  principally  with  the  external 
relations  of  bodies  and  their  action  upon  one  another.  If,  how- 
ever, the  bodies  are  so  large  as  to  constitute  worlds,  of  which  the 
earth  itself  is  one,  this  science  takes  the  name  of  Astronomy. 

The  word  Physics  is  much  used  to  include  both  Natural 
Philosophy  and  Chemistry;  but  sometimes  it  is  applied  to  the 
branches  of  Natural  Philosophy,  except  Mechanics.  According  to 
this  use  of  the  word,  Natural  Philosophy  is  divided  into  two  gen- 
eral subjects,  Mechanics  and  Physics. 

2.  Definitions  relating  to  Matter. — 

A  body  is  a  separate  portion  of  matter,  whether  large  or 
small. 

An  atom  is  a  portion  of  matter  so  small  as  to  be  indivisible. 

A  particle  denotes  the  smallest  portion  which  can  result 
from  division  by  mechanical  means,  and  consists  of  many  atoms 
united  together. 

The  word  molecule  signifies  a  very  small  portion  of  matter, 
either  atom  or  particle. 

Mass  is  the  quantity  of  matter  in  a  body,  and  is  usually 
measured  by  its  weight. 

Volume  signifies  the  space  occupied  by  a  body. 

Density  expresses  the  relative  mass  contained  within  a  given 


MECHANICS. 

*  Ttolume.    \T,hug,;if  :oiie  body  has  twice  as  great  a  mass  within  a  cer- 
i' vohtnte  aV  another  has,  it  is  said  to  have  twice  the  density. 

\tirb  *thb  ^inufce  portions  of  space  within  the  volume  of 
i  which'  cares  not  filled  by  the  material  of  that  body.    All 
matter  is  porous,  some  kinds  in  a  greater  and  some  in  a  less  degree. 
Force  is  the  name  of  any  cause,  whatever  it  may  be,  which 
gives  motion  to  matter,  or  which  changes  its  motion. 

3.  Properties  of  Matter. — 

(1.)  Extension. — Every  portion  of  matter,  however  small,  has 
length,  breadth,  and  thickness,  and  thus  occupies  space.  This 
is  its  extension. 

(2.)  Impenetrability. — While  matter  occupies  space,  it.  ex- 
cludes all  other  matter  from  it,  so  that  no  two  atoms  can  be  in 
exactly  the  same  place  at  the  same  time.  This  property  is  called 
impenetrability. 

The  two  foregoing  are  often  called  essential  properties,  because 
we  cannot  conceive  matter  to  exist  without  them. 

(3.)  Divisibility. — Matter  is  divisible  beyond  any  known  limits. 
After  being  divided,  as  far  as  possible,  into  particles  by  mechani- 
cal methods,  it  may  be  still  further  reduced  by  chemical  action  to 
atoms,  which  are  too  small  to  be  in  any  way  recognized  by  the 
senses, 

(4:.)  Compressibility. — Since  pores  exist  in  all  matter,  it  may 
be  compressed  into  a  smaller  volume.  Hence  all  matter  is  com- 
pressible, though  in  very  different  degrees. 

(5.)  Elasticity. — After  a  body  has  suffered  compression,  it 
shows,  in  some  degree  at  least,  a  tendency  to  restore  itself  to  its 
former  volume.  This  property  is  called  elasticity.  A  body  is  said 
to  be  perfectly  elastic  when  the  force  by  which  it  recovers  its  size 
is  equal  to  that  by  which  it  was  before  compressed.  The  word 
elasticity  is  used  generally  in  a  wider  sense  than  is  given  in  the 
above  definition,  namely,  the  tendency  which  a  body  has  to  recover 
its  original  form,  whatever  change  of  form  it  may  have  previously 
received.  Thus,  if  a  body  is  stretched,  bent,  twisted,  or  distorted 
in  any  other  way,  it  is  called  elastic,  if  it  tends  to  resume  its  form 
as  soon  as  the  force  which  altered  it  has  ceased.  Torsion  is  the 
name  of  the  elastic  force  which  tends  to  untwist  a  thread  or  wire 
when  it  has  been  twisted. 

(6.)  Attraction. — This  is  the  general  name  used  to  express  the 
universal  tendency  of  one  portion  of  matter  towards  another.  It 
receives  different  names,  according  to  the  circumstances  in  which 
it  acts.  The  attraction  which  binds  together  atoms  of  different 
kinds,  so  as  to  form  a  new  substance,  is  called  affinity,  and  is  dis- 
cussed in  Chemistry ;  that  which  unites  particles,  whether  simple 


INTRODUCTION.  3 

or  compound,  so  as  to  form  a  body,  is  called  cohesion;  the  cling- 
ing of  two  kinds  of  matter  to  each  other,  without  forming  a  new 
substance,  is  called  adhesion;  and  the  tendency  manifested  by 
masses  of  matter  toward  each  other,  when  at  sensible  distances,  is 
called  gravity. 

'(7.)  Inertia. — This  is  also  a  universal  property  of  matter,  and 
signifies  its  tendency  to  continue  in  its  present  condition  as  to 
motion  or  rest.  If  at  rest,  it  cannot  move  itself;  if  in  motion,  it 
cannot  stop  itself  or  change  its  motion,  either  in  respect  to  direc- 
tion or  velocity. 

4.  Branches  of  Natural  Philosophy.— Natural  Philosophy 
is  generally  divided  into  Mechanics,  Hydrostatics,  Pneumatics, 
Sound,  Magnetism,  Electricity,  Heat,  and  Light. 

Mechanics  treats  of  the  motion  and  equilibrium  of  bodies3 
caused  by  the  application  of  force.  Since-  there  are  three  condi- 
tions of  matter,  solid,  liquid,  and  gaseous,  it  is  convenient  to 
divide  the  general  subject  of  Mechanics  into  three  branches. 

1st.  The  mechanics  of  solids,  also  called  Mechanics. 

2d.  The  mechanics  of  liquids,  called  Hydrostatics. 

3d.  The  mechanics  of  gases,  called  Pneumatics. 

All  the  other  branches  of  Natural  Philosophy  (often  called 
Physics)  treat  of  various  phenomena  caused  by  minute'  vibrations 
in  the  particles  of  matter.  These  vibrations  are  excited  in  differ- 
ent ways,  and  when  transmitted  to  us,  affect  one  or  more  of  our 
senses.  Thus,  sound  consists  of  such  vibrations  as  affect  the  sense 
of  hearing ;  and  light  is  another  mode  of  vibration,  that  affects 
only  the  sense  of  vision. 

It  was  formerly  customary  to  regard  magnetism,  electricity, 
heat,  and  light,  as  so  many  kinds  of  imponderable  matter,  that  is, 
matter  having  no  sensible  weight,  and  thus  distinguished  from 
solids,  liquids,  and  gases,  which  are  the  different  forms  of  ponder- 
able matter.  But  it  is  now  known  that  when  forces  are  applied  to 
matter,  they  not  only  produce  the  visible  forms  of  motion,  but 
may  be  made  to  develop  either  sound,  magnetism,  electricity,  heat, 
or  light;  and  that  most  of  these  modes  of  motion  may  be  trans- 
formed into  others,  and  each  may  be  made  a  measure  of  the  force 
which  is  employed  to  produce  it. 


PART  I. 

MECHANICS 


CHAPTER  I. 

MOTION    AND    FORCE. 

5.  Classification  of  Motions. — Motion  is  change  of  place, 
and  is  either  uniform  or  variable.    In  uniform  motion  equal 
spaces  are  passed  over  in  equal  times,  however  small  the  times  may 
be.     In  variable  motion  the  spaces  described  in  equal  times  are  un- 
equal.   Such  motion  may  be  either  accelerated  or  retarded.    In 
accelerated  motion  the  spaces  described  in  equal  times  become  con- 
tinually greater;  in  retarded  motion  they  become  continually  less. 
Motion  is  said  to  be  uniformly  accelerated  if  the  increments  of 
space  in  equal  times  (however  small)  are  equal ;  and  uniformly  re- 
tarded if  the  decrements  are  equal. 

Velocity  is  the  space  described  in  the  unit  of  time.  In  Me- 
chanics, one  second  is  much  used  as  the  unit  of  time,  and  one  foot 
as  the  unit  of  space ;  hence,  velocity  is  the  number  of  feet  de- 
scribed in  one  second. 

6.  Uniform  Motion. — When  motion  is  uniform,  the  number 
of  feet  described  in  one  second,  multiplied  by  the  number  of 
seconds,  obviously  gives  the  whole  space.    Let  s  =  space,  t  =  time, 

o  o 

and  v  =  velocity;   then  s  =  t  v;  .'.  t  —  -,  and  v  —  -.    If  this 

space  is  compared  with  another,  s',  described  in  the  time  t1,  with 
the  velocity  v'9  then  s  :  s' : :  tv  :  t'  v' ;  or  briefly,  in  the  form  of  a 

o  o 

variation,  s  oc  t  v.    In  like  manner  t  cc  -,  and  v  oc  -. 

v  t 

If  two  bodies,  moving  uniformly,  describe  equal  spaces,  then 
s  =  s' ;  .*.  t  v  =  t'  v' ;  .'.  t  \  t' : :  v'  :v.  That  is,  in  order  that  two 
bodies  may  describe  equal  spaces,  their  velocities  must  vary  in- 
versely as  the  times  during  which  they  move. 

7.  Questions  on  Uniform  Motion. — 

1.  A  ball  was  rolled  oh  the  ice  with  a  velocity  of  78  feet  per 
second,  and  moved  uniformly  21  seconds ;  what  space  did  it  de- 
scribe? Ans.  1638  feet. 


MOMENTUM.  5 

2.  A  steamboat  moved  uniformly  across  a  lake  17  miles  wide, 
at  the  rate  of  20  feet  per  second ;  what  time  was  occupied  in 
crossing?  Ans.  "Lh.  14m.  4s. 

3.  On  the  supposition  that  the  earth  describes  an  orbit  of  600 
millions  of  miles  in  365^  days,  with  what  velocity  does  it  move  per 
second?  A ns.  19  miles,  nearly. 

4.  Three  planets  describe  orbits  which  are  to  each  other  as  15 
19,  and  12,  in  times  which  are  as  7,  3,  and  5 ;  what  are  their  rela- 
tive velocities?  Ans.  225,  665.  and  252. 

8.  Momentum. — The  momentum  of  a  body  signifies  its  quan- 
tity of  motion,  and  is  reckoned  according  to  the  mass,  or  quantity 
of  matter,  which  is  moving,  and  the  velocity  with  which  it  moves. 
The  momentum,  therefore,  varies  as  the  product  of  the  mass  and 
the  velocity. 

Let  the  momentum  of  a  body  =  m,  its  mass  =  q,  and  its  velo- 

m  m 

city  =  v\  then  m  =  q  v,  q  =  — ,  and  v  =  — .  In  order  to  com- 
pare the  momentum  of  one  body  with  that  of  another,  let  m',  q',  v', 
represent  the  momentum,  mass,  and  velocity,  respectively  of  the 

77? 

second  body;   then  mi  m' : :  q  v  :  q'  v';  or  m  oc  q  v;   .•.  q  cc  — , 

,         m 
and  v  cc  — . 
2 

If  the  momentum  of  one  body  equals  that  of  another,  then, 
since  m  —  m',  q  v  =  q'  v',  :.  q  :  q' : :  v' :  v.  That  is,  in  order  that 
the  momenta  of  two  bodies  should  be  equal,  their  masses  must 
vary  inversely  as  their  velocities. 

Since  there  are  two  elements  entering  into  the  momentum  of  a 
body — namely,  its  mass,  usually  expressed  in  pounds,  and  its  velo- 
city, expressed  in  feet  per  second — therefore  momentum  cannot  be 
measured  either  in  pounds  or  in  feet,  being  in  nature  unlike  either. 
The  word,  foot-pound  is  employed  for  the  unit  of  momentum  when- 
ever the  unit  of  mass  is  a  pound  and  the  unit  of  velocity  is  a  foot 
per  second.  £ 

9.  Questions  on  Momentum. —  /£'**<• 

1.  A  ship  weighing  336,000  Ibs.  is  dashed  against  the  rocks  in  &MH 
a  storm,  with  a  velocity  of  16  miles  per  hour ;  with  what  momen-    <**** 
turn  did  she  strike  ?  Ans.  7,884,800  foot-pounds.      >H  *** 

2.  A  ball  weighing  1  oz.  is  fired  into  a  log  weighing  53  Ibs., 
suspended  so  as  to  move  freely,  and  imparts  a  velocity  of  2  ft.  per 
second.    Assuming  that  the  log  and  ball  have  a  momentum  equal 
to  the  previous  momentum  of  the  ball  alone,  required  the  velocity 
of  the  ball.  Ans.  1,698  ft.  per  sec. 

3.  Suppose  a  comet,  whose  velocity  is  1,000,000  miles  per  hour, 


6  MECHANICS. 

has  the  same  momentum  as  the  earth,  whose  velocity  is  19  miles 
per  second ;  what  is  the  ratio  of  their  masses  ?         Ans.  1 : 14.6. 

4.  Two  railway  cars  have  their  quantities  of  matter  as  7  to  3, 
and  their  momenta  as  8  to  5 ;  what  are  their  relative  velocities  ? 

Ans.  As  24  to  35,  or  nearly  5  to  7. 

5.  The  momentum  of  a  cannon-ball  was  434  foot-pounds ;  what 
must  be  the  velocity  of  a  half-ounce  bullet,  in  order  to  have  the 
same  momentum  ?  Ans.  13,888  feet. 

10.  Classification  of  Forces. — The  principal  forces  in  na- 
ture are  the  following: 

1.  Attraction  in  its  several  forms.     Cohesion  and  chemical  af- 
finity are  the  forces  which  bind  together  the  particles  and  atoms 
of  bodies,  and  gravity  is  that  which  everywhere  near  the  earth 
causes  bodies  to  fall  toward  it,  or  to  press  upon  it. 

2.  Elasticity. — This  is  a  force  which,  in  many  kinds  and  con- 
ditions of  matter,  tends  to  repel  the  particles  from  each  other. 

The  forces,  whether  attraction  or  repulsion,  which  exist  among 
the  atoms  or  molecules  of  a  body,  are  called  molecular  forces. 

3.  Muscular  force. — All  living  beings  are  endowed  with  this 
force,  by  which  they  put  in  motion  bodies  around  them,  and  by 
acting  upon  other  bodies,  are  enabled  also  to  move  themselves 
from  place  to  place. 

4.  Matter  in  motion. — If  a  body  which  some  force  has  put  in 
motion  impinges  on  another  body,  it  imparts  motion  to  it,  and  is 
therefore  itself  a  force.    This  is  true  not  only  of  ordinary  visible 
motions,  but  of  those  small  and  often  invisible  vibrations,  which 
manifest  themselves  as  sound,  heat,  &c.    Gravity,  or  any  other 
force,  may  cause  heat,  and  heat  may  cause  light  and  electricity. 
Thus,  any  form  of  motion  is  a  force,  and  it  can  be  employed  to 
produce  other  forms. 

11.  Impulsive  and  Continued  Forces  and  their  Ef- 
fects.— An  impulsive  force  is  one  which  has  no  sensible  continu- 
ance, as  the  blow  of  a  hammer.    A  continued  force  is  one  which 
acts  during  a  perceptible  length  of  time.    Continued  forces  are 
subdivided  into  constant  and  variable.    A  constant  force  has  the 
same  intensity  during  the  whole  time  of  its  action;    a  variable 
force  is  one  whose  intensity  changes. 

Keeping  in  mind  the  property  of  inertia,  we  associate  different 
kinds  of  motion  with  the  forces  which  produce  them,  as  follows : 

1.  An  impulsive  force  causes  uniform  motion. 

2.  A  continued  force,  accelerated  motion. 

3.  A  constant  force,  uniformly  accelerated  motion. 

4.  A  variable  force,  unequally  accelerated  motion. 

If  the  force  is  applied  in  a  direction  opposite  to  that  in  which 


MOTION    AND    FORCE.  7 

the  body  has  a  previous  uniform  motion,  the  conn  action  is  the 
following  : 

5.  An  impulsive  force  causes  uniform  motion,  or  rest. 

6.  A  continued  force,  retarded  motion. 

7.  A  constant  force,  uniformly  retarded  motion. 

8.  A  variable  force,  unequally  retarded  motion. 

In  cases  1  and  5,  it  is  obvious  that,  the  impulse  being  given, 
the  body  is  left  to  itself,  and  cannot  change  the  state  of  motion  or 
rest  impressed  on  it 

In  2,  3,  and  4,  it  must  be  considered  that  the  force  at  each  in- 
stant adds  a  new  increment  to  the  uniform  motion  which  the  body 
would  have  had  if  the  force  had  ceased;  and  if  the  force  is 
constant,  those  increments  are  equal;  if  variable,  they  are  un- 
equal. 

In  6,  7,  and  8,  the  same  statements  may  be  made  in  regard  to 
decrements.  It  is  also  plain  that  in  these  three  last  cases,  if  the 
force  continues  to  act  indefinitely,  the  motion  will  be  retarded 
until  the  body  comes  to  a  state  of  momentary  rest,  and  then  is 
accelerated  in  the  direction  of  the  force. 

12.  Measure  of  Force. — The  intensity  of  an  impulsive  force 
is  measured  by  the  momentum  which  it  will  produce  or  destroy ; 
that  is,  /  oc  m.    But  m  x  q  v ;  /.  /  oc  q  v.    Hence,  if  q  is  con- 
stant, /  QC  v.    If,  then,  an  impulse  is  applied  to  a  given  mass,  the 
intensity  of  that  impulse  is  measured  by  the  velocity  which  it  im- 
parts or  destroys. 

But  in  the  case  of  a  constant  force,  the  momentum  depends  not 
only  on  the  intensity  of  the  force,  but  on  the  time  during  which  it 

is  applied;  that  is,/£  oc  m,  and/  oc  — .    If  the  mass  of  the  body 

t 

is  given,  then,  as  in  the  case  of  an  impulsive  force,  q  being  con- 

v 

stant,  ftccv,  and  /  oc  -. 
t 

To  express  the  measure  of  a  variable  force,  let  t  be  a  constant 
and  infinitely  small  portion  of  time  ;  then  the  force  varies  as  the 
mass  multiplied  by  the  increment  of  velocity  imparted  in  that 
time. 

13.  The  Three  Laws  of  Motion.— All  the  phenomena  of 
motion  in  Mechanics  and  Astronomy  are  found  to  be  in  accord- 
ance with  three  first  principles,  which  Newton  announced  in  his 
Principia,  and  which  are  to  be  regarded  as  forming  the  basis  of 
mechanical  science.    They  may  be  named  and  defined  as  follows : 

1.  The  law  of  inertia. — A  body  at  rest  tends  to  remain  at  rest ; 
and  a  body  in  motion  tends  to  move  forever,  in  a  straight  line,  and 
uniformly. 


g  MECHANICS. 

2.  The  law  of  the  coexistence  of  motions. — If  several  motions 
are  communicated  to  a  body,  it  will  ultimately  be  in  the  same 
position,  whether  those  motions  are  simultaneous  or  successive. 

3.  The  law  of  action  and  reaction. — If  any  kind  of  action  takes 
place  between  two  bodies,  it  produces  equal  momenta  in  opposite 
directions;  or,  every  action  is  accompanied  by  an  equal  and  oppo- 
site reaction. 

The  truth  of  these  laws  cannot  be  established,  except  approxi- 
mately, by  direct  experiments,  because  gravity,  friction,  and  the 
resistance 'of  air,  interfere  more  or  less  with  every  possible  experi- 
ment They  are  to  be  learned  rather  by  a  careful  study  of  the 
phenomena  of  motion  in  general.  We  see  an  approximation  to  the 
first  law,  in  rolling  a  ball  on  a  horizontal  surface ;  first,  on  the 
earth,  then  on  a  floor,  and  again  on  smooth  ice,  the  motion  ap- 
proaching toward  uniformity  as  obstructions  are  diminished,  and 
gravity  producing  no  direct  effect,  because  acting  at  right  angles 
to  the  line  of  motion.  The  discussion  of  the  second  law  is  reserved 
for  Chapter  III.  The  third  law  is  illustrated  by  a  variety  of  cases 
in  collision,  attraction,  and  repulsion.  Suppose  that  a  body  A, 
being  in  motion,  strikes  directly  against  B,  which  is  at  rest ;  it  is 
found  that  B  acquires  a  certain  momentum,  and  that  A  loses  (that 
is,  acquires  in  an  opposite  direction)  an  equal  amount.  The  same 
is  true  if  B  is  in  motion,  and  A  either  overtakes  or  meets  it.  In 
the  collision  of  two  railroad  trains,  it  is  immaterial  as  to  the 
effects  which  they  will  respectively  suffer,  whether  each  is  moving 
towards  the  other,  or  whether  one  is  at  rest,  provided  that  in  the 
latter  case  the  moving  train  has  a  momentum  equal  to  the  mo- 
menta of  the  two  trains  in  the  former  case.  When  a  magnet 
attracts  a  piece  of  iron,  each  moves  towards  the  other  with  the 
same  momentum.  A  spring  between  two  bodies  A  and  B  drives 
A  from  B  with  as  much  momentum  as  B  from  A  ;  and  the  sudden 
expansion  of  burning  gunpowder,  which  propels  the  balls  when  a 
broadside  is  fired,  causes  an  equal  amount  of  motion  of  the  ship  in 
the  opposite  direction. 

14.  Force  of  Gravity. — Every  mass  of  matter  near  the 
earth,  when  free  to  move,  pursues  a  straight  line  towards  its 
centre.  The  force  by  which  this  motion  is  produced  is  called 
gravity  ;  either  the  gravity  of  the  body  or  the  gravity  of  the  earth ; 
for  the  attraction  is  mutual  and  equal,  in  accordance  with  the 
third  law  of  motion.  It  is  easy  to  understand  why  a  small  mass 
should  attract  a  large  one,  as  much  as  the  large  mass  attracts  the 
small  one.  Let  A  consist  of  one  atom  of  matter,  and  B,  at  any 
distance  from  it,  consist  of  ten  atoms.  If  it  be  admitted  that  A 
attracts  one  atom  of  B  as  much  as  that  one  atom  attracts  A,  then 


FORCE    OF    GRAVITY.  9 

the  above  conclusion  follows.  For  A  attracts  each  of  the  ten 
atoms  of  B  as  much  as  each  of  the  same  ten  attracts  A  ;  so  that  A 
exerts  ten  units  of  attraction  on  B,  while  B  exerts  ten  units  of 
attraction  on  A.  The  same  reasoning  obviously  applies  to  the 
earth  in  relation  to  the  small  bodies  on  its  surface. 

15.  Relation  of  Gravity  and  Mass.  —  At  the  same  dis- 
tance from  the  centre  of  the  earth,  gravity  varies  as  the  mass. 
This  is  because  it  operates  equally  on  every  atom  of  a  body  ;  hence 
the  greater  the  number  of  atoms  in  a  body,  the  greater  in  the  same 
ratio  is  the  attraction  exerted  upon  it.     That  gravity  varies  as  the 
mass  is  also  proved  from  the  observed  fact,  that  in  a  vacuum  it 
gives  the  same  velocity,  in  the  same  time,  to  every  mass,  however 
great  or  small,  and  of  whatever  species  of  matter.     For  a  constant 
force,  acting  for  a  given  time,  is  measured  by  the  momentum  which 
it  produces  (Art.  12),  and  that  momentum,  if  the  velocity  is  the 
same,  varies  as  the  mass  :  therefore  the  force  also  varies  as  the 
mass  to  which  it  imparts  the  given  velocity. 

If  a  body  is  not  free  to  move,  its  tendency  towards  the  earth 
causes  pressure  ;  and  the  measure  of  this  pressure  is  called  the 
weight  of  the  body.  Weight  is  usually  employed  as  a  measure  of 
the  mass  in  bodies.  The  foregoing  relations  are  embodied  in  the 
following  expressions  :  g  <x  q  ;  and  w  oc  q. 

16.  Relation  of  Gravity  and  Distance.  —  At  different  dis- 
tances from  the  earth,  gravity  varies  inversely  as  the  square  of  the 
distance  from  the  centre.    The  demonstration  of  this  proposition  is 
reserved  for  astronomy,  where  it  is  shown  by  the  movements  of 
the  bodies  in  the  solar  system  that  this  law  applies  to  them  all. 

The  moon  is  60  times  as  far  from  the  earth's  centre  as  the  dis- 
tance from  that  centre  to  the  surface  :  therefore  the  attraction  of 
the  earth  upon  the  particles  of  the  moon  is  3600  times  less  than 
upon  particles  at  the  surface  of  the  earth.  At  the  height  of  4000 
miles  above  the  earth,  gravity  is  four  times  less  than  at  the  surface. 
But  the  heights  at  which  experiments  are  commonly  made  upon 
the  weights  of  bodies  bear  so  small  a  ratio  to  the  radius  of  the 
earth,  that  this  variation  is  commonly  imperceptible.  At  the 
height  of  half  a  mile,  the  diminution  does  not  amount  to  more 
than  about  ^^th  Par^  of  the  weight  at  the  surface.  For,  let 
r  =  the  radius  of  the  earth  =  4000  miles,  nearly  ;  and  let  x  be  the 
height  of  the  body,  w  its  weight  at  the  earth's  surface,  and  w'  its 
weight  at  the  height  x.  Then, 

w.w':\  (r  +  x)*  :  r2  :  : 


But  when  x  is  a  small  fraction  of  r,  x*  may  be  neglected,  and 


10  MECHANICS. 

the  formula  becomes  w  —  w'  = 


„        •    •    .  .  •     •     •       •• 

Let  z  be  half  a  mile;  then  .QQQ_,  -.   =  IIHJT^  Par*  °f  tne  whole 

weight  ;  or,  a  body  would  weigh  so  much  less  at  the  height  of  half 
a  mile  than  at  the  surface  of  the  earth.  But  if  the  height  were  as 
great  as  100  miles  above  the  earth,  the  loss  should  be  calculated 
by  formula  (A),  since  the  other  would  give  a  result  too  small  by 
one  per  cent,  or  more,  according  to  the  height. 

What  loss  of  weight  would  a  body  sustain  by  being  elevated  500 
miles  above  the  earth  ?  Ans.  |f,  or  more  than  \  of  its  weight. 

The  relation  of  gravity  to  distance  is  expressed  by  the  formula 

g  oc  —  ;  and  as  g  oc  q  also,  it  varies  as  the  product  of  the  two  ; 
cL 

that  is,  g  <x  ~  ;  or  gravity  towards  the  earth  varies  as  the  mass  of 

the  body  directly,  and  as  the  square  of  the  distance  from  the  earth's 
centre  inversely. 

17,  Gravity  within  a  Hollow  Sphere.  —  A  particle  situated 
within  a  spherical  shell  of  uniform  density,  is  equally  attracted  in 
all  directions,  and  remains  at  rest.  This  is  true,  because,  in  every 
direction  from  the  body,  the  mass  varies  at  the  same  rate  as  the 
square  of  the  distance,  so  that  attraction  increases  for  one  reason, 
as  much  as  it  diminishes  for  the  other  ;  which  is  proved  as  follows  : 

Let  the  particle  P  (Fig.  1)  be  at  any  point 
within  the  spherical  shell  A  B  CD.    Let  two 
opposite  cones  of   revolution,  of   very  small 
angle,  have  their  vertices  at  P,  and  suppose  the 
figure  to  be  a  section  through  the  centre  of  the 
sphere  and  the  axis  of  the  cones.    Then  A  B 
and  a  ~b  will  be  the  major  axes  of  the  small 
ellipses,  which  are  the  bases  of  the  cones,  and 
which  may  be  considered  as  plane  figures.     By 
geometry,  A  P  :  P  B  :  :  P  I  :  P  a:,   and  the  angles  at  P  being 
equal,  the  triangles  are  similar  ;    hence  the  angles  B  and  a  arc 
equal.    Therefore,  the  bases  of  the  cones  are  similar  ellipses,  being 
sections  of  similar  cones,  equally  inclined  to  the  sides.    By  similar 
triangles,  A  P*  :  P  V  :  A  B*  :  a  b\     Let  q  and  q  represent  the 
masses  of  the  thin  laminas  which  form  the  bases;  then,  since  sim- 
ilar ellipses  are  to  each  other  as  the  squares  of  their  major  axes,  we 

have  ,      •  A  D3     „  ,2  q  q' 

qiq'-.'.AP'iPV,  or  -fpi  =  ?y. 

But  -~^  and  —  ^-  represent  the  attractions  of  the  bases  respec- 
A  1  ±  o 


FORCE    OF    GRAVITY.  H 

tively  on  the  particle  (Art.  16) ;  and  since  these  are  equal,  the 
particle  is  equally  attracted  by  all  the  opposite  parts  of  the  spheri- 
cal shell. 

18.  Gravity  "within  a  Solid  Sphere. — Within  a  solid  sphere 
of  uniform  density,  weight  varies  directly  as  the  distance  from  the 
centre. 

Let  a  particle  P  (Fig.  2)  be  within  the  solid  FIG.  2. 

sphere  ADO;  and  call  its  distance  from  the 
centre  d.  Now,  by  the  preceding  article  the 
shell  exterior  to  it,  ADR,  exerts  no  influence 
upon  it,  and  it  is  attracted  only  by  the  sphere 
P  R  8.  Let  q  represent  the  quantity  of  this 

sphere ;  then  gravity  varies  as  ~.    But  q  oc  d3 ; 

.-.gcc-^azd.    Hence,  in  the  earth  (if  it  be  supposed  spherical 

and  uniformly  dense,  though  it  is  neither  exactly),  a  body  at  the 
depth  of  1000  miles  weighs  three-fourths  as  much  as  at  the  surface, 
and  at  2000  miles  it  weighs  half  as  much,  while  at  the  centre  it 
weighs  nothing. 

Comparing  this  proposition  with  Art.  16,  we  learn  that  just  at 
the  surface  of  the  earth  a  body  weighs  more  than  at  any  other 
place  without  or  within.  Within,  the  weight  diminishes  nearly  as 
the  distance  from  the  centre  diminishes ;  without,  it  diminishes  as 
the  square  of  the  distance  from  the  centre  increases. 

At  the  surface  of  spheres  having  the  same  density,  weight  varies 
as  the  radius  of  the  sphere.  Let  r  be  the  radius  of  the  sphere,  and 

a  r3 

q  its  mass ;  then,  since  g  oc  \,  in  this  case  it  varies  as  —  oc  r. 

Therefore,  if  two  planets  have  equal  densities,  the  weight  of  bodies 
upon  them  is  as  their  radii  or  their  diameters.  If  a  ball  two  feet 
in  diameter  has  the  same  density  as  the  earth,  a  particle  of  dust  at 
its  surface  is  attracted  by  it  nearly  21  millions  of  times  less  than  it 
is  by  the  earth. 

19.  Questions  for  Practice. — 

1.  How  much  weight  would  a  rock  that  weighs  ten  tons 
(22,400  Ibs.)  at  the  level  of  the  sea,  lose  if  elevated  to  the  top  of  a 
mountain  five  miles  high?  Ans.  55.8952  Ibs. 

2.  If  the  earth  were  a  hollow  sphere,  and  if,  through  a  hole 
bored  through  the  centre,  a  man  were  let  down  by  a  rope,  would 
the  force  required  to  support  him  be  increased  or  diminished  as 
he  descended  through  the  solid  crust,  and  where  would  it  become 
equal  to  nothing  ? 

3.  How  much  would  a  44-pound  shot  weigh  at  the  centre  of 


12  MECHANICS. 

the  earth ;  how  much  at  a  point  half  way  from  the  centre  to  the 
surface ;  and  how  much  100  miles  below  the  surface  ? 

4.  If  a  hole  were  bored  through  the  centre  of  the  earth,  and  a 
stone  were  dropped  into  it,  in  what  manner  would  the  stone  move 
in  its  way  to  the  centre  and  after  it  reached  the  centre  ? 

^*  Suppose  a  32-pound  cannon-ball,  fired  with  the  velocity  of 
2,000  foot  per  second,  to  have  the  same  momentum  as  a  battering- 
ram  whose  weight  is  5760  pounds;  find  the  velocity  of  the  latter. 

Ans'  1L11  ft'  Per  sec' 
to  have  weight,  and  one  grain  of  it  moving  at 

ra^e  °f  192,000  miles  per  second,  to  impinge  directly  against  a 
mass  of  ice  moving  at  the  rate  of  1.45  feet  per  second,  and  to  stop 
it ;  requires  the  weight  of  the  ice. 

Ans.  99877.832  Ibs.,  or  nearly  44|  tons,  reckoning  7000  gr.=l  Ib. 

7.  If  a  ball  of  the  same  density  with  the  earth,  T\,th  of  a  mile 
diameter,  were  to  fall  through  its  own  diameter  toward  the 

earth,,what  space  would  the  earth  move  through  to  meet  the  ball, 
Siameter  of  the  earth  being  taken  at  8,000  miles  ? 
Tt  Am.  Hooo<jJoooo<j  inch,  nearly. 

8.  Two  men  are  pulling  a  boat  ashore  by  a  rope,  one  at  each 
end,  A  being  in  the  boat  and  B  on  the  shore ;  how  will  the  time 
of  bringing  the  boat  ashore  compare  with  the  time  in  which  A 
would  pull  it  ashore  alone,  were  the  other  end  of  the  rope  fixed  to 
an  immovable  post  ? 

9.  Suppose  the  rope  to  pass  from  A  in  one  boat  to  B  in 
another  equal  boat ;  how  fast  will  B's  boat  move  ?  will  A's  boat 
have  the  same  velocity  as  when  B  was  on  the  shore  ? 


CHAPTER   II. 

VARIABLE  MOTION.    FALLING  BODIES. 

20.  Uniform  Motion  represented  G-eometrically. — When 
a  body  moves  uniformly  for  a  given  time,  the  space  described 
equals  the  time  multiplied  by  the  velocity 
(Art.  6).    Therefore,  if  one  side  of  a  rect- 
angle  represents  the  time  of  motion,  and 
an  adjacent  side  the  uniform  velocity,  the 
area  of  the  rectangle  will  represent  the 
space  described  in  that  time,  because  the 
area  equals  the  product  of  two  adjacent 
sides.    Thus,  let  A  B,  B  C,  &c.  (Fig.  3),   D 
represent  any  equal  portions  of  time,  and 
A  Fy  B  6r,  &c.,  the  uniform  velocity; 


ACCELERATED    MOTION. 


13 


FIG.  4. 


H 


N 


O 


then  A  G,  B  H,  &c.,  may  be  used  to  represent  the  spaces  de- 
scribed, and  the  rectangle  A  L  may  represent  the  space  passed 
over  with  the  velocity  A  F,  in  the  time  A  E. 

21.  Velocity  Increased  at  Finite  Intervals. — Suppose 
the  body  to  receive  equal  impulses  at  the  beginning  of  all  the 
equal  portions  of  time,  A  B,  B  C,  &c.  (Fig.  4).    Then,  A  F  being 
the  velocity  given  by  the  first  im- 
pulse, G  H,  K  L,  &c.,  the  increments  A 

of  velocity,  will  each  be  equal  to  A  F\ 
and  B  H,  the  velocity  during  the  B 
second  portion  of  time,  equals  2  A  F\ 
CL,  that  of  the  third,  equals  3  A  F,  &c.  c 
Therefore,  B  G,  C  K,  D  M,  &c.,  are 
as  3,  2,  3,  etc.  But  A  B,  A  C,  A  D,  D 
&c.,  are  as  1,  2,  3,  &c.  Hence  the  tri- 
angles A  B  G,  A  C  K,  &c.,  are  simi- 
lar,  and  the  same  straight  line  A  0 
passes  through  the  angles  of  all  the  rectangles.  Now  these  rect- 
angles represent  the  successive  spaces  described  in  the  equal 
times,  and  their  sum  represents  the  whole  space  described  in  the 
time  A  E.  This  exceeds  the  triangle  A  0  E  by  the  sum  of  the 
small  equal  triangles  A  F  G,  G  H  K,  &c. 

22.  Uniformly  Accelerated  Motion  Represented. — Let 

the  increments  of  time  and  velocity  (Fig.  5)  be  half  as  great  as 
before.   Then  the  sum  of  all  the  rect- 
angles, or  the  whole  space  described, 
exceeds  the  triangle  A  0  E  by  the 
sum  of  the  triangles  A  f  g,  g  F  G,  &c. 
These  triangles  are  one-half  the  sum 
of  those  in  Fig.  4.    Therefore,  by  con-   C 
tinually  halving  the  increments  of 
time  and  velocity,  the  sum  of  the  rect-  D 
angles    continually    approaches  the 
area  of  the  triangle ;  and  when  these 


FIG.  5. 


increments  become  infinitely  small, 

the  first  velocity  becomes  zero,  and  the  sum  of  the  rectangles 

equals  the  triangle.     Therefore,  the  space  described  by  a  body 

which  begins  to  move  from  rest  by  the  action  of  a  constant  force 

may  be  represented  by  a  right-angled  triangle,  as  A  0  E,  whose 

side  A  E  represents  the  time,  and  the  side  E  0  the  last  acquired 

velocity. 

Such  motion  is  said  to  be  uniformly  accelerated  (Art.  11).  An 
example  of  this  is  found  in  the  fall  of  a  body  in  a  vacuum.  For 
gravity  acts  incessantly,  and  within  the  range  of  our  experiments 


14 


MECHANICS. 


it  may  be  considered  as  acting  with  equal  intensity.  The  proper- 
ties of  the  triangle  enable  us  to  ascertain  very  readily  the  laws  of 
the  fall  of  a  body. 

23.  Laws  of  the  Fall  of  Bodies.—  When  bodies  fall  from 
rest  by  the  force  of  gravity,  and  unobstructed  by  the  air,  the  fol- 
lowing relations  exist  between  the  space,  time,  and  velocity: 

1.  The  spaces  vary  as  the  squares  of  the  times. 

2.  The  spaces  vary  as  the  squares  of  the  acquired  velocities. 

3.  The  times  vary  as  the  acquired  velocities. 

For  let  s  be  the  space  described,  v  the  velocity  acquired  by  a 
body  falling  from  rest  for  the  time  t,  sr  the  space  described,  v'  the 
velocity  acquired  at  any  other  period  t'  of  its  fall  ;  then,  from  what 
has  already  been  demonstrated,  if  t  and  t'  be  represented  by  the 
lines  A  B  and  A  D  (Fig.  6),  and  v  and  v'  by  the  lines  B  C  and 
D  E,  drawn  at  right  angles  to  them,  s  and  s1  will  be  FIG.  6. 
represented  by  the  triangles  A  B  C,  A  D  E.  Now, 


hence,  s  :  s'  :  :  f  :  tn,  or  as  v*  :  vr<*. 

As  equal  increments  of  velocity  are  generated  in 
equal  times,  it  is  farther  evident  that  the  velocity 
acquired  varies  as  the  time  ;  the  same  conclusion 
may  also  be  deduced  'from  the  similar  triangles 
A  B  O,A  DE-,  forB  C:D  E::A  B:AD,i.e.v 


Since  the  spaces  described  are  as  the  squares  of  the  times,  if 
a  body  falls  from  rest  during  times  which  are  represented  by  the 
numbers  1,  2,  3,  4,  5,  &c.,  the  spaces  described  in  those  times  will 
be  as  the  square  numbers  1,  4,  9,  16,  25,  &c.;  and  the  spaces  de- 
scribed in  equal  successive  portions  of  time  will  be  as  the^  odd 
numbers  1,  3,  5,  7,  9,  &c.,  as  exhibited  in  the  following  table  : 


Times. 

Spaces  described. 

Spaces  described  in  equal  successive  portions  of  time. 

I 
2 

3 
4 

a! 

I 
4 

£ 

£ 

In  ist  portion  of  time 
2d 
3d        " 
4th 
5th       «          « 
&c  

i 

.     .     .    4-    1  =  3 
...    9-    4  =  5 
.     .     .  16  -    9  =  7 
.    .    .  25  —  16  =  9 
.     .    .  &c.        —  &c. 

The  odd  number  expressing  the  space  described  in  any  unit 
of  time,  and  which  is  found  in  the  above  table  by  taking  the  differ- 
ence of  the  squares  of  successive  numbers,  may  also  be  obtained  by 
subtracting  one  from  twice  the  number  of  units  in  the  time. 
Thus,  in  the  table,  3  =  2x2- 1;  5  =  2x3  —  1,  &c.  This 
is  true  to  any  extent.  Let  n  represent  any  whole  number ;  the 


RETARDED    MOTION.  15 

number  next  less  is  n  —  1.  The  space  described  in  n  units  of 
time  is  represented  by  ?&2,  and  that  described  in  n  —  1  units,  by 
(n  —  I)2.  Therefore,  the  space  described  in  the  nth  unit  is  repre- 
sented by  ri*  —  (n  —  1)"  =  2  n  —  1.  This  is  an  odd  number, 
and  it  equals  twice  the  given  number,  less  one. 

24.  Uniformly  Retarded  Motion.— If  a  body  be  projected 
perpendicularly  upward  in  a  vacuum,  with  the  velocity  which  it 
has  acquired  in  falling  from  any  height,  it  will  rise  to  the  point 
from  which  it  fell,  before  it  begins  to  descend  again,  and  the  mo- 
tion will  be  uniformly  retarded.    As  the  force  of  gravity  adds 
equal  velocities  in  equal  times  to  a  descending  body,  so  it  destroys 
equal  velocities  in  equal  times  in  a  body  which  is  ascending.    The 
spaces  described  in  successive  units  of  time,  by  a  body  thus  ascend- 
ing, reckoning  from  the  beginning  of  its  motion,  will  be  the  same 
as  those  stated  in  the  foregoing  table,  but  in  an  inverted  order : 
thus,  if  the  time  be  divided  into  four  equal  parts,  then  the  spaces 
described  in  the  descent  of  the  body  during  these  equal  times  are 
as  the  numbers  1,  3,  5,  7,  but  in  its  ascent  they  will  be  as  7, 5, 3, 1 ; 
that  is,  the  space  described  in  the  first  portion  of  time,  in  its 
ascent,  will  be  the  same  as  that  described  in  the  last,  in  its  de- 
scent, and  so  on  till  the  body  arrives  at  its  highest  point. 

25.  Acquired  Velocity.— If  a  body  moves  uniformly  with 
the  acquired  velocity,  it  will  pass  over  twice  as  great  a  space,  in 
the  same  time,  as  it  falls  through  to  acquire  it. 

Let  the  triangle  ABC  (Fig.  7)  represent  the  Af\  FlG-  7- 
space  described  by  gravity  in  the  time  A  B,  and 
B  C  the  last  acquired  velocity ;  produce  A  B  to 
D,  making  B  D  equal  to  A  B,  and  complete  the 
rectangle  B  E\  then,  if  a  body  moves  during  the 
time  B  D  with  the  uniform  velocity  represented 
by  B  C,  the  space  described  in  that  time  will  be 
represented  by  the  rectangle  B  E\  but  the  tri- 
angle A  B  C  is  half  B  E\  hence  the  space  de- 
scribed with  the  velocity  B  C  continued  uniformly 
is  twice  that  which  would  be  described  in  the'  same  time  A  B, 
falling  from  rest. 

Since  the  space  described  by  a  body  falling  from  rest  is  half 
that  which  it  would  describe  in  the  same  time  with  its  greatest 
Telocity  continued  uniformly,  and  since  a  body  projected  per- 
pendicularly upward  rises  to  the  same  height  as  that  from  which 
it  must  fall  to  acquire  the  velocity  of  projection,  the  whole  space 
described  by  a  body  projected  perpendicularly  upward  is  half  that 
which  it  would  describe  in  the  same  time  with  its  first  velocity 
continued  uniformly. 


16 


MECHANICS. 


FIG.  8. 


26.  Projection  Downward.  —  The  space  described  in  any 
time  by  a  body  projected  downward  with  a  given  velocity  is  equal 
to  the  space  which  would  be  described  with  that  velocity  continued 
uniformly  during  that  time,  together  with  the  space  through  which 
a  body  would  fall  from  rest  by  the  action  of  gravity  in  the  same  time. 

Let  A  D  (Fig.  8)  represent  the  given  velocity  of  projection,  and 
A  B  the  given  time,  and  complete  the  rectangle 
A  E\  produce  B  E  to  (7,  and  let  E  G  represent 
the  velocity  generated  by  gravity  in  the  time  A  B 
or  D  E,  and  join  D  O.  Then  the  body,  moving 
by  projection  alone  (Art.  20),  would  describe  the 
rectangle  A  E  in  the  time  A  B\  but,  by  gravity 
alone,  it  would  describe  the  triangle  DEC  (Art. 
22).  Hence,  by  the  coexistence  of  both  motions 
(Art.  13),  it  would  describe  the  trapezoid  A  C. 

27.  Projection  Upward.  —  The  space  described  by  a  body 
ascending  for  a  given  time  is  equal  to  the  space  described  uni- 
formly with  the  velocity  of  projection  in  that  time,  diminished  by 
the  space  fallen  through  from  rest  in  the  same  time. 

Let  B  C  (Fig.  9)  be  the  velocity  of  projection,  and  A  B  the 
time  in  which  a  body  would  acquire  that  velocity 
in  falling  from  rest.  Then  the  triangle  ABC  repre- 
sents the  space  through  which  it  would  ascend  before 
the  velocity  is  lost.    Let  B  E  be  the  given  time  of 
ascent;  then  the  rectangle  B  D  is  the  space  de- 
scribed in  the  time  B  E,  with  the  velocity  B  C  con- 
tinued uniformly,  and  CDF  (similar  to  A  B  C)  the 
space  fallen  through  in  the  same  time.  But  the  part 
B  E  F  C  of  the  triangle  A  B  C  is  the  space  through  which  the 
body  ascends  in  the  time  B  E;  and  this  is  equal  to  the  difference 
of  the  rectangle  B  D  and  the  triangle  CDF. 

28.  Formulae  for   the  Fall   of  Bodies.—  The  distance 
through  which  a  body  falls  in  a  vacuum  in  one  second  of  time 
varies  on  different  parts  of  the  earth.     Between  latitudes  40°  and 
50°,  it  is  very  nearly  16^  feet,  or  193  inches.   Therefore  (Art.  25), 
at  the  end  of  the  second  the  body  is  moving  with  a  velocity  which, 
if  gravity  were  to  cease,  would  carry  it  over  32  J  feet  per  second. 
Let  g  =  321  feet,  the  velocity  acquired  in  one  second  of  fall.  Then 
\g  —  !6T1o,  the  distance  of  fall  in  the  first  second.    Let  s  be  the 
space  described,  and  v  the  velocity  acquired,  in  any  other  time  t. 
Then,  according  to  the  laws  of  variation  (Art.  23),  we  have  : 

(1.)  ft;ii;.:£;ifl.iy  .............  s  =  4  g1\ 


FlG 


(2.) 


and 


FALLING    BODIES. 


17 


(3.)  * : 
(4.)  . 


and 


(6.) 


and 


FIG.  10. 


29.  Space  and  Time  Represented*  by  Co-ordinates.— 

The  relation  of  space  and  time  in  different  kinds  of  motion  may 
be  well  represented  by  the  rectangular  co-ordinates  of  certain  lines.- 
Thus,  in  uniform  motion  we  have 

s  =  v  t, 

in  which  v  is  constant.  This  may  be 
regarded  as  the  equation  of  a  straight 
line  passing  through  the  origin,  and 
making  with  the  axis  of  abscissas  an 
angle,  whose  tangent  is  v.  Therefore, 
if  any  abscissa  A  C  (Fig.  10)  repre- 
sents the  number  of  units  of  time  oc- 
cupied in  the  motion,  the  corresponding  ordinate  C  D  will  repre- 
sent the  space  passed  over. 

Again,  for  the  uniformly  accelerated  motion  of  a  falling  body 
we  have 

s  =  %  g  F9  or  P  =  -  s. 

*/ 

This  is  the  equation  of  a  parabola  whose  parameter  is  - 

y 

Therefore,  if  the  parabola  A  B  (Fig.  11)  be  described,  having 
-  for  its  parameter,  and  the  time  of  fall- 

0 

ing  is  represented  by  any  ordinate  C  D, 
the  corresponding  abscissa  A  G  will 
represent  the  space  fallen  through. 

This  is  illustrated  by  Morin's  appa- 
ratus, where  a  body  falls  parallel  to  the 
axis  of  a  uniformly  revolving  cylinder, 
wrapped  with  paper,  against  which  a 
pencil,  attached  to  the  falling  body, 
gently  presses.  When  the  paper  is  un- 
wound and  developed  upon  a  plane,  the  curve  traced  by  the  pencil 
is  found  to  be  a  parabola. 

30.  Applications  of  Formulae  for  the  Fall  of  Bodies.— 

1.  A  body  falls  6  seconds ;  what  space  does  it  pass  over,  and 
what  velocity  does  it  acquire  ?     Ans.  s— 579  ft.  v= 193  ft.  per  sec. 
2 


18  MECHANICS. 

2.  How  far  must  a  body  fall  to  acquire  a  velocity  of  50  feet  per 
second,  and  how  long  will  it  be  in  falling  ? 

Am.  s  =  38.86  ft.  t  =  1.55  sec. 

3.  A  body  fell  from  the  top  of  a  tower  150  feet  high ;  liow  long 
was  it  in  falling,  and  what  velocity  did  it  have  at  the  bottom  ? 

Am.  t  =  3.054  sec.  v  =  98.237  ft. 

4.  If  a  ball  be  thrown  upward  with  a  velocity  of  100  feet  per 
ttt'b    second,  what  height  will  it  reach  ?  Am.  155.44  ft 

5.  Suppose  a  body  to  fall  during  3  seconds,  and  then  to  move 
.uniformly  during  2  seconds  more,  with  the  velocity  acquired; 
what  is  the  whole  distance  passed  over? 

The  space  fallen  through  is  16^  x  9  —  144|  feet.  The  velo- 
city acquired  is  32|  x  3  =  96J  feet.  The  space  described  uni- 
formly is  96-J  x  2  =  193  feet.  Therefore  the  whole  space  is 
144}  +  193  =  337|  feet- 

•  6.  A  ball  fired  perpendicularly  upward  was  gone  10  seconds, 
when  it  returned  to  the  same  place ;  how  high  did  it  rise,  and  with 
what  velocity  was  it  projected?  Am.  $=402^  ft.,  v=160|  ft 

31.  Space  in  any  Given  Second  or  Seconds  of  Fall- 
Since  the  spaces  described  in  the  successive  units  of  time  are  as 
the  odd  numbers,  and  as  -J  g  is  described  in  the  first  second,  there- 
fore 3  x  £  g  is  described  in  the  second,  5  x  £  g  in  the  third,  and 
generally  (2  n  —  I)  x  \g  in  the  nth  second. 

1.  How  far  does  a  body  move  in  the  14th  second  of  its  fall? 

Am.  434|  ft 

2.  A  body  had  been  falling  2  minutes;  how  far  did  it  move  in 
the  last  second?  Am.  3843} £  feet 

The  space  described  in  the  last  m  seconds  is  found  thus :  The 
space  in  the  whole  time  t,  =  \  g  f ;  and  in  the  time  t  —  m,  the 
space  —  |  g  (t  —  m)2.  Subtracting  the  latter  from  the  former,  we 
find  the  space  described  in  the  last  m  seconds  to  be  %g(2mt  —  m*). 
When  m  =  1,  this  becomes  for  the  space  in  the  last  second  ^  g 
(2  t  —  1).  This  is  the  same  form  of  expression  as  was  found 
above,  where  n  was  a  whole  number  of  seconds.  Therefore,  the 
space  described  in  any  one  second  of  the  fall,  whether  the  time 
from  the  beginning  is  an  integral  or  a  fractional  number,  is  found 
by  multiplying  4  g  by  twice  the  number  of  seconds  minus  one. 

3.  What  space  was  described  in  the  last  two  seconds  by  a  body 
which  had  fallen  300  feet?  Am.  213.58  feet 

4.  A  body  had  been  falling  8J  seconds ;  how  far  did  it  descend 
in  the  next  second  ?  Am.  289^  ft 

32.  Calculation  for  Projection  Upward  or  Downward. — 

A  body  projected  downward  describes  t  v  feet  by  the  force  of 'pro- 
jection, and  £  g  f  feet  by  the  force  of  gravity  (Art.  26).    A  body 


FALLING    BODIES.  19 

projected  upward  describes  tv  by  the  force  of  projection ;  but  this 
is  diminished  by  A  g  f,  which  gravity  would  cause  it  to  describe  in 
the  same  time  (Art.  27).  Therefore  the  formula  for  space  de- 
scribed by  a  body  projected  downward  is  t  v  +  $gt*;  by  a  body 
projected  upward,  the  formula  is  t  v  —  ^  g  t* . 

1.  A  body  is  projected  downward  with  a  velocity  of  30  feet  in  a 
second;  how  far  will  it  fall  in  4  seconds?  Am.  377  J  ft 

2.  A  body  is  projected  upward  with  a  velocity  of  120  feet  in  a 
second;  how  far  will  it  rise  in  3  seconds?          .      Ans.  215|  ft 

3.  Suppose  at  the  same  instant  that  a  body  begins  to  fall  from 
rest  from  the  point  D  (Fig.  12),  another  body  is  projected 
upward  from  B  with  a  velocity  which  would  carry  it  to  A  ;  FIG.  12. 
it  is  required  to  find  the  point  where  they  would  meet. 

Let  0  be  the  point  where  the  bodies  would  meet ;  and 
let  A  B  =  a,  B  D  —  b,  D  G  =  x ;  then  will  A  D  =  a — #, 
A  G  —  a  —  #  +  #v  j 

Now  the  time  of  descending  through  D  C  =  I — )  ;  and 
the  time  of  ascending  through  B  C  (—  time  down  A  B  — 
time  down  ,4  0)  =  (— )*-  fl  (a  ~  *>  +  *$.  but  the  time 

down  D  0  must  be  equal  to  the  time  up  B  G\  hence  we  have 
/2jpU_  /2  flU     /2 (a  - 1  4-  x\j        i_    i  ^ 

\9  '    ~W  '       \         £         /  '°] 

.*.  (#  —  5  +  #)2:=  #2 —  #2, and#  — - 5  4-  x  =  a+x— 2 (##)¥; 

/.  2  (#  #)^~  =  J,  or  4  a  #  =  52,  and  a;  =  — . 

4.  Suppose  a  body  to  have  fallen  from  A  to  B  (Fig.  13),  when 
another  body  begins  to  fall  from  rest  at  D\  how  far  will  the  latter 
body  fall  before  it  is  overtaken  by  the  former? 

Let  C  be  the  point  where  one  body  overtakes  the  other,  FIG.  13. 

Now  time  down  D  C  =  I— )*,  and  time  down  B  C  =  time 

i  .1 

down  AC-  time  down  A  B  =(2  ^  +  l  +  ^\-  /— )2; 

V  #  )       \9  ' 

but  at  the,  moment  when  'the  lower  body  is  overtaken,  time 
down  D  C  =  time  down  B  C,  or 

<« "    _  ' 

.D 


33.  Questions  on  Falling  Bodies.— 

1.  The  momentum  of  a  meteoric  stone  at  the  instant  of 


20  MECHANICS. 

striking  the  earth  was  estimated  at  18435  foot-pounds,  and  it  had 
been  falling  10  seconds ;  from  what  height  did  it  fall,  and  what 
was  its  weight?  Ans.  1608J  ft;  57.31  Ibs. 

2.  An  archer  wishing  to  know  the  height  of  a  tower,  found 
_-X    that  an  arrow  sent  to  the  top  of  it  occupied  8  seconds  in  going  and 

returning;  what  was  the  height  of  the  tower?         Ans.  257^  ft. 

3.  In  what  time  would  a  man  fall  from  a  balloon  three  miles 
nigh,  and  what  velocity  would  he  acquire  ? 

Ans.  t  =  31.38  sec.;  v  =  1009.39  ft. 

4.  A  body  having  fallen  for  3J  seconds,  was  afterwards  observed 
to  move  with  the  velocity  which  it  had  acquired  for  2|  seconds 
more ;  what  was  the  whole  space  described  by  the  body  ? 

Ans.  478|  ft.,  very  nearly. 

5.  Through  what  space  would  the  aeronaut ~(in  Question  3)  fall 
during  the  last  second  ?  Ans.  993.3  feet. 

6.  A  body  has  fallen  from  the  top  of  a  tower  340  feet  high ; 
what  was  the  space  described  by  it  in  the  last  three  seconds  ? 

Ans.  298.957  ft. 

7.  Suppose  a  body  be  projected  downward  with  a  velocity  of  18 
feet  in  a  second ;  how  far  will  it  descend  in  15  seconds  ? 

Ans.  3888|  ft. 

8.  A  body  is  projected  upward  with  a  velocity  of  65  feet  in  a 
second ;  how  far  will  it  rise  in  two  seconds  ?  Ans.  65|  ft. 

9.  "With  what  velocity  must  a  stone  be  projected  into  a  well 
450  feet  deep,  that  it  may  arrive  at  the  bottom  in  four  seconds  ? 

Ans.  48^  ft.  in  a  second. 

10.  The  space  described  in  the  fourth  second  of  fall  was  to  the 
space  described  in  the  last  second  except  four,  as  1 :  3 ;  what  was 
the  whole  space  described  by  the  body  ?  Ans.  3618|  ft. 

11.  A  staging  is  at  the  height  of  84  feet  above  the  earth.    A 
ball  thrown  upward  from  the  earth,  after  an  absence  of  7  seconds, 
fell  on  the  staging ;  what  was  the  velocity  of  projection  ? 

Ans.  124.58  ft.  per  second. 

12.  A  body  is  projected  upward  with  a  velocity  of  483  feet  in  a 
second;  in  what  time  will  it  rise  to  a  height  of  1610  feet? 

Ans.  t  —  3.82  sea,  or  26.2  sec. 

13.  From  a  point  214f  feet  above  the  earth  a  body  is  projected 
'  upward  with  a  velocity  of  161  feet  in  a  second ;  in  what  time  will 

it  reach  the  surface  of  the  earth,  and  with  what  velocity  will  it 
strike  ?  Ans.  t  —  11.2  sec.,  v  —  199  ft. 

14.  Suppose  a  body  to  have  fallen  through  50  feet,  when  a 
second  body  begins  to  fall  just  100  feet  below  it ;  how  far  will 
the  latter  body  fall  before  it  is  overtaken  by  the  former  ? 

Ans.  50  ft. 

15.  A  body  is  projected  upward  with  a  velocity  of  64|  feet  in  a 


ATWOOD'S    MACHINE. 


FIG.  14. 


second;  how  far  above  the  point  of  projection  will  it  be  at  the  end. 
of  4  seconds  ?  Am.  0  ft. 

16.  A  body  is  projected  upward  with  a  velocity  of  128f  feet  in 
a  second;  where  will  it  be  at  the  end  of  10  seconds  ? 

Ans.  321f  ft.  below  the  point  of  projection. 

[See  Appendix  for*  the  discussion  of  the  fall  of  bodies  by  the 
Calculus.] 

34.  Atwood's  Machine. — Accurate  observations  on  the  di- 
rect fall  of  a  body  cannot  be  easily 
made,  on  account  of  its  great  velocity ; 
and  if  they  could  be,  the  relations  be- 
tween time,  space,  and  acquired  velo- 
city, would  not  be  found  to  agree  with 
those  obtained  by  calculation,  on  ac- 
count of  the  resistance  of  the  air.  Ex- 
periments on  falling  bodies  are  usually 
performed  by  an  instrument  known  as 
Atwood's  machine,  represented  in  Fig. 
14.  From  the  base  of  the  instrument, 
which  is  furnished  with  leveling  screws, 
rises  a  substantial  pillar,  about  seven 
feet  high,  supporting  a  small  table  upon 
the  top. 

Above  the  table  is  a  grooved  wheel, 
delicately  suspended  on  friction-wheels, 
and  protected  from  dust  by  a  glass  case. 
Two  equal  poises,  M  and  M',  are  at- 
tached to  the  ends  of  a  fine  cord,  which 
passes  over  the  groove  of  the  wheel.  As 
gravity  exerts  equal  forces  on  M  and 
M'9  they  are  in  equilibrium.  To  set 
them  in  motion,  a  small  bar  m  is  placed 
on  J/,  which  will  immediately  begin  to 
descend,  and  M'  to  rise.  But  this  mo- 
tion will  be  slower  than  in  falling  freely, 
because  the  force  which  gravity  exerts 
on  the  bar  must  be  communicated  to 
the  poises,  and  also  to  the  revolving 
wheel  over  which  the  cord  passes.  By 
increasing  the  poises  M,  M',  and  dimin- 
ishing the  bar  m,  the  motion  may  be 
made  as  slow  as  we  please.  H  is  a 
simple  clock  attached  to  the  pillar  for  J 
measuring  seconds,  and  for  dropping  f| 
the  poise  M  at  the  beginning  of  a  vibra-  ^  ,^^ 


22  MECHANICS. 

tion  of  the  pendulum.  Q  is  a  scale  of  inches  extending  from  the 
base  to  the  table.  The  stage  A  may  be  clamped  to  any  part  of  the 
scale,  in  order  to  stop  the  poise  M  in  its  descent,  as  represented  at 
C.  The  ring  B,  which  is  large  enough  to  allow  the  poise,  but  not 
the  bar,  to  pass  through  it,  is  also  clamped  to  the  scale  wherever 
the  acceleration  is  to  cease. 

Let  M  be  raised  to  the  top,  and  held  in  place  by  a  support,  and 
then  let  the  pendulum  be  set  vibrating.  When  the  index  passes 
the  zero  point,  the  clock  causes  the  support  to  drop  away,  and  the 
poise  descends.  The  pendulum  shows  how  many  seconds  elapse 
before  the  bar  is  arrested  by  the  ring,  and  how  many  more  before 
the  poise  strikes  the  stage.  From  the  top  to  the  ring  the  motion 
is  accelerated  by  the  constant  fraction  of  gravity  acting  on  it ; 
from  the  ring  to  the  stage  the  poise  moves  uniformly  with  the 
acquired  velocity.  All  the  formulse  relating  to  the  fall  of  a  body 
can  therefore  be  illustrated  by  these  slow  motions.  Moreover,  the 
resistance  of  the  air  is  so  much  diminished  when  the  motion  is 
slow,  that  a  good  degree  of  correspondence  is  found  to  exist  be- 
tween the  experiments  and  the  results  of  calculation. 

35.  Living  Force. — We  have  seen  that  when  a  body  is  pro- 
jected upward  in  a  vacuum,  the  height  to  which  it  will  rise  varies 
as  the  square  of  the  velocity  of  projection.  In  the  ascent,  the 
body  is  constantly  and  uniformly  opposed  by  gravity.  If  the 
motion  of  a  given  body  were  opposed  by  any  other  uniform  ob- 
struction, the  distance  it  would  proceed  before  coming  to  rest 
would  also  vary  as  the  square  of  the  velocity.  This  power  to  over- 
come a  constant  resistance  varies,  therefore,  not  as  the  mo- 
mentum— that  is,  as  the  product  of  mass  and  velocity  q  v — but  as 
the  product  of  mass  and  square  of  velocity  q  v* ;  and  it  is  called 
the  vis  viva  or  living  force,  in  distinction  from  vis  inertice  or  dead 
force.  If  a  ball  weighing  one  pound  move  with  the  velocity  of 
2,000  feet,  and  another  ball  weighing  two  pounds  move  with  the 
velocity  of  1,000  feet,  then  the  momentum  (q  v)  of  the  first  equals 
that  of  the  second.  But  the  living  force  (q  v2)  of  the  first  is  twice 
as  great  as  that  of  the  second ;  for  1  x  20008 :  2  x  10002 : :  2  :  1. 
The  first  body,  in  its  ascent,  will  reach  four  times  as  great  a  height 
as  the  second ;  or,  if  the  two  balls  be  fired  into  a  bank  of  earth, 
the  first  will  penetrate  four  times  as  far  as  the  second. 

In  practical  mechanics,  the  living  force  is  generally  called  the 
working  power  ;  and  the  work  which  it  will  perform  is  therefore 
measured  by  the  mass  multiplied  by  the  square  of  the  velocity. 
The  working  power  of  the  steam  in  a  locomotive  is  employed  in 
maintaining  a  certain  velocity  in  the  train,  in  spite  of  grade,  fric- 
tion of  rails  and  machinery,  and  resistance  of  the  air.  If  the  power 


LIVING    FORCE.  33 

of  the  steam  were  wholly  cut  off,  the  train  would  be  uniformly  re- 
tarded by  the  constant  resistances,  until,  after  running  a  certain 
distance,  it  would  come  to  rest.  But  if  the  velocity  of  the  train 
had  been  twice  as  great,  it  would  have  run  four  times  as  far  before 
stopping ;  it  would  also  have  required  four  times  as  great  a  force 
to  give  the  train  this  double  velocity.  Both  of  these  facts,  one  re- 
lating to  the  effect,  the  other  to  the  cause,  show  that  the  working 
power  is  to  be  estimated  according  to  the  square  of  the  velocity. 
So  the  working  power  employed  in  moving  any  kind  of  machinery, 
which  presents  a  constant  resistance,  varies  as  the  square  of  the 
velocity  imparted,  and  the  work  performed  by  the  machinery  is 
reckoned  in  the  same  way.  To  give  a  missile  greater  velocity  is 
more  advantageous  than  to  increase  its  mass.  A  40-pound  ball, 
with  1400  feet  velocity,  is  7  times  more  efficient  in  penetrating 
the  walls  of  forts  and  the  hulks  of  ships  than  a  280-pound  ball  with 
200  feet  velocity,  though  the  momentum  is  the  same  in  each  case. 

36.  Measure  of  Force. — In  Art.  12  force  is  said  to  be 
measured  by  momentum,  or/  oc  q  v;  and  in  Art.  35  it  is  said  to  be 
measured  by  the  work  performed,  or  /  oc  q  v".  But  these  state- 
ments are  not  to  be  considered  as  inconsistent  with  each  other ; 
for  in  the  first  case,  force  has  reference  to  inertia  ;  in  the  second 
case  it  has  reference  to  work.  "When  a  force  acts  on  a  body  that  is 
free  to  move  without  obstruction  (which  is,  however,  only  a  sup- 
posable  case),  the  effect  is  perpetual;  the  body  will  move  on 
uniformly  forever.  If  *the  force  had  been  greater,  the  velocity 
would  have  been  greater  in  the  same  ratio.  But  when  resistances 
oppose  (as  is  always  true  in  practice),  then  the  force  is  expended 
in  overcoming  them,  and  this  is  the  ivork  to  be  performed ;  and  if 
the  force  ceases  to  operate,  the  motion  will  at  length  cease  also ; 
but,  as  has  been  shown,  the  space  passed  over,  and  therefore  the 
work  performed,  will  vary  as  the  square  of  the  velocity. 

When  force  is  employed  to  perform  work,  it  is  by  some  writers 
called  energy,  to  distinguish  it  from  force  as  used  in  producing 
momentum.  Hence  energy  varies  as  the  product  of  the  mass  and 
the  square  of  the  velocity. 


CHAPTER   III. 

COMPOSITION  AND  RESOLUTION  OF  MOTION. 

37.  Motion  by  Two  or  More  Forces. — Motion  produced 
by  a  single  force,  either  impulsive  or  continued,  has  been  already 
considered.  But  motion  is  more  generally  caused  by  several  forces 
acting  in  different  directions. 


24  MECHANICS. 

When  two  or  more  forces  act  at  once  on  a  body,  each  force  is 
called  a  component,  and  the  joint  effect  is  called  the  resultant. 
Forces  may  be  represented  by  the  straight  lines  along  which  they 
would  move  a  body  in  a  given  time ;  the  lines  represent  the  forces 
in  two  particulars,  the  directions  in  which  they  act  and  their  rela- 
tive magnitudes.  Whenever  an  arrow-head  is  placed  on  a  line,  it 
shows  in  which  of  the  two  directions  along  that  line  the  force  acts. 

38.  The  Parallelogram  of  Forces. — This  is  the  name 
given  to  the  relation  which  exists  betweeen  any  two  components 
and  their  resultant,  and  is  stated  as  follows : 

If  two  forces  acting  at  once  on  a  lody  are  represented  by  the 
adjacent  sides  of  a  parallelogram)  their  resultant  is  expressed  ~by  the 
diagonal  which  passes  through  the  intersection  of  those  sides. 

Suppose  that  a  body  situated  at  A  (Fig.  15)  receives  an  impulse 
which,  acting  alone,  would  carry  it 
over  A  B  in  a  given  time,  and  an-  A 

other  which  would  carry  it  over  A  0 
in  the  same  length  of  time.  If  both 
impulses  are  given  at  the  same  in- 
stant, the  body  describes  A  D  in  the 
same  time  as  A  B  by  the  first  force, 
or  A  C  by  the  second,  and  the  motion  in  A  D  is  uniform. 

This  is  an  instance  of  the  coexistence  of  motions,  stated  in  the 
second  law  of  motion  (Art  13).  For  the  body,  in  passing  directly 
from  A  to  D,  is  making  progress  in  the  direction  A  0  as  rapidly 
as  though  the  force  A  B  did  not  exist ;  and  at  the  same  time  it 
advances  in  the  direction  A  B  as  fast  as  though  that  were  the  only 
force.  When  the  body  reaches  D,  it  is  as  far  from  the  line  A  B  as 
if  it  had  passed  over  A  (7;  it  is  also  as  far  from  the  line  A  C  as  if 
it  had  gone  over  A  B.  Thus  it  appears  that  both  motions  A  B 
and  A  C  fully  coexist  in  the  progress  of  the  body  along  the  diag- 
onal A  D.  That  the  motion  is  uniform  in  the  diagonal  is  evi- 
dent from  the  law  of  inertia ;  for  the  body  is  not  acted  on  after  it 
leaves  A. 

It  is  evident  that  a  single  force  might  produce  the  same  effect ; 
that  force  would  be  represented,  both  in  direction  and  magnitude, 
by  the  line  A  D.  The  force  A  D  is  said  to  be  equivalent  to  the 
two  forces  A  B  and  A  C. 

39.  Velocities  Represented. — The  lines  A  B  and  A  C  are 
described  by  the  components  separately,  and  the  line  A  D  by  their 
joint  action,  in  the  same  length  of  time.    Hence  the  velocities  in 
those  lines  are  as  the  lines  themselves.    In  the  parallelogram  of 
forces,  therefore,  two  adjacent  sides  and  the  diagonal  between  them 
represent — 


TRIANGLE    OF    FORCES.  35 

1st.  The  directions  of  the  components  and  resultant ; 

2d.  Their  relative  magnitudes ;  and 

3d.  The  relative  velocities  with  which  the  lines  are  described. 

40.  The  Triangle  of  Forces. — For  purposes  of  calculation, 
it  is  more  convenient  to  represent  two  components  and  their  re- 
sultant by  the  sides  of  a  triangle,  than  by  the  sides  and  diagonal 
of  a  parallelogram.    In  Fig.  15,  C  D,  which  is  equal  and  parallel  to 
A  B,  may  represent  in  direction  and  magnitude  the  same  force 
which  A  B  represents.    Therefore,  the  components  are  A  C  and 
CD,  while  their  resultant  is  A  D ;  and  the  angle  C  in  the  triangle 
is  the  supplement  of  GAB,  the  angle  between  the  components. 
Care  should  be  taken  to*  construct  the  triangle  so  that  the  sides 
representing  the  components  may  be  taken  in  succession  in  the 
directions  of  the  forces,  as,  A  C,  CD;  then  A  D  correctly  repre- 
sents their  resultant.    But,  although  A  C  and  A  B  represent  the 
components,  the  third  side,  C  B,  of  the  triangle  A  C  B,  does  not 
represent  their  resultant,  since  A  C  and  A  B  cannot  be  taken  suc- 
cessively in  the  direction  of  the  forces.    It  is  necessary  to  go  back 
to  A  in  order  to  trace  the  line  A  B.    It  should  be  observed,  that 
though  C  D  represents  the  magnitude  and  direction  of  the  compo- 
nent, it  is  not  in  the  line  of  its  action,  because  both  forces  act 
through  the  same  point  A. 

41.  The  Forces  Represented  Trigonometrically.— Since 

the  sides  of  a  triangle  are  proportional  to  the  sines  of  the  opposite 
angles,  these  sines  may  also  represent  two  components  and  their 
resultant.  Thus,  the  sine  ofOAD  corresponds  to  the  component 
A  B  (—  CD) ;  the  sine  of  CD  A  (=  D  A  B)  corresponds  to  the 
component  A  C;  and  the  sine  of  C  (=  sine  of  C  A  B}  corresponds 
to  the  resultant  A  D.  Each  of  the  three  forces  is  therefore  repre- 
sented by  the  sine  of  the  angle  between  the  other  two. 

42.  Greatest  and  Least  Values  of  the  Resultant— A 

change  in  the  angle  between  the  components  alters  the  value  of 
the  resultant;  as  the  angle  increases  from  0°  to  180°,  the  resultant 
diminishes  from  the  sum  of 
the  components  to  their  differ-  FlG- 

ence.  In  Fig.  16,  let  CAB 
and  D  A  B  be  two  different 
angles  between  the  same  com- 
ponents A  C  (or  A  D}  and 
A  B.  As  C  A  B  is  less  than 
DAB,  its  supplement  A  B  F 
is  greater  than  A  B  E,  the  A  JB 

supplement  of  DA  B',  there- 
fore A  F  is  greater  than  A  E.  When  the  angle  C  A  B  is  dimin- 


MECHANICS. 


FIG.  17. 


islied  to  0°,  tlie  sides  A  B,  B  F,  become  one  straight  line,  and  A  F 
equals  their  sum ;  when  D  A  B  is  enlarged  to  180°,  E  falls  on 
A  B,  and  A  E  equals  the  difference  of  A  B  and  A  C.  Between 
the  sum  and  difference  of  the  components,  the  resultant  may  have 
all  possible  values. 

43.  The  Polygon  of  Forces.— All  the  sides  of  a  polygon 
except  one  may  represent  so  many  forces  acting  at  the  same  time 
on  a  body,  and  the  remaining  side  will  represent  their  resultant. 
In  Fig.  17,  suppose  A  B,  A  C, 

and  A  D,  to  represent  three 
forces  acting  together  on  a  body 
at  A.  The  resultant  of  A  B  and 
A  C  is  represented  by  the  diag- 
onal A  E\  and  the  resultant 
of  AE  and  A  D  by  the  diagonal 
A  F.  As  A  F  is  equivalent  to 
A  E  and  A  D,  and  A  E  is  equiv- 
alent to  A  B  and  A  C,  therefore 
A  F  is  equivalent  to  the  three,  A  B,  A  C,  and  A  D.  But  if  we 
substitute  B  E  for  A  0,  and  E  F  for  A  D,  then  the  three  compo- 
nents are  A  B,  B  E,  and  E  F,  three  sides  of  a  polygon,  and  the 
resultant  A  F  is  the  fourth  side  of  the  same  polygon. 

So,  in  Fig.  18,  A  B,  B  C,  C  D,  D  E,  and  E  F,  may  represent 
the  directions  and  relative  magni- 
tudes of  five  forces,  which  act  simul- 
taneously on  a  body  at  A.  The  re- 
sultant of  A  B  and  B  C  is  A  (7;  the 
resultant  of  A  C  and  C  D  is  A  D ; 
the  resultant  of  A  D  and  D  E  is 
A  E',  and  the  resultant  of  A  E  and 
JBFis  A  F;  which  last  is  therefore 
the  resultant  of  all  the  forces,  A  B, 
B  C,  CD,  DE,  and^  F\  the  com- 
ponents being  represented  by  five 
sides,  and  their  resultant  by  the  sixth  side,  of  a  polygon  of  six 
sides. 

44.  Curvilinear  Motion. — Since,  according  to  the  first  law 
of  motion,  a  moving  body  proceeds  in  a  straight  line,  if  no  force 
disturbs  it,  whenever  we  find  a  body  describing  a  curve,  it  is  cer- 
tain that  some  force  is  continually  deflecting  it  from  a  straight 
line.    Besides   the   original   impulse,  .therefore,   which    gave  it 
motion  in  one  direction,  it  is  subject  to  the  action  of  a  continued 

•  force,  which  operates  in  another  direction.  A  familiar  example 
occurs  in  the  path  of  a  projectile.  Suppose  a  body  to  be  thrown 


FIG.  18. 


CALCULATION  OF  RESULTANT.        27 

from  P  (Fig.  19),  with  an  impulse  which  would  alone  carry  it  to 

N9  in  the  same  time  in  which  gravity  alone  would  carry  it  to  V. 

Complete  the  parallelogram  P  Q ;  FIG  19 

then,  as  both  motions  coexist  (2d 

law),  the  body  at  the  end  of  the 

time  will  be  found  at  Q.    Let  t  be 

the  time  of  describing  P  Nor  P  V\ 

and  let  tf  be  the  time  of  describing 

P  M  by  the  impulse,  or  P  L  by 

gravity.    Then,  at  the  end  of  the 

time  t',  the  body  will  be  at  0.   Now, 

as  P  N  is  described  uniformly, 

PN\PM\'.t'.t'  ',.\PN*:PM>:: 

f :  £'a. 

But  (Art. 23), P  F:  P  £ : :  f :  T; 
.-.  P  V:  P  L  : :  P  N* :  P  M*;  or  Q  V  :  0  L\ 

Hence,  the  curve  is  such  that  P  Foe  Q  F8 ;  that  is,  the  abscissa 
varies  as  the  square  of  the  ordinate,  which  is  a  property  of  the 
parabola.  P  0  0  is  therefore  a  parabola,  one  of  whose  diameters 

OF2 
is  P  F,  and  the  parameter  to  that  diameter  is  ^-= 

Owing  to  the  resistance  of  the  air,  the  curve  deviates  sensibly 
from  a  parabola,  especially  in  swift  motions. 

45.  Calculation  of  the  Resultant  of  Two  Impulsive 
Forces. — When  two  components  and  the  angle  between  them  are 
given,  the  resultant  may  be  found  both  in  direction  and  magni- 
tude by  trigonometry.  The  theorem  required  is  that  for  solving  a 
triangle,  when  two  sides  and  the  included  angle  are  given ;  but  the 
included  angle  is  not  that  between  the  components,  but  its  supple- 
ment (Art.  40).  In  Fig.  15,  if  A  B  =  54,  and  A  C=  22,  and 
CAB  —  75°,  then  A  C  D  is  the  triangle  for  solution,  in  which 
A  0=  22,  CD  =  54,  and  A  0 D  =  105°.  Performing  the  cal- 
culation, we  find  the  resultant  A  D  =  63.363,  and  the  angle  D  A  B, 
which  it  makes  with  the  greater  force,  =  19°  35'  43".  This 
method  will  apply  in  all  cases. 

1.  A  foot-ball  received  two  blows  at  the  same  instant,  one  directly 
east,  at  the  rate  of  71  feet  per  second,  the  other  exactly  northwest, 
at  the  rate  of  48  feet  per  second ;  in  what  direction  and  with  what 
velocity  did  it  move  ?  Ans.  1ST.  47°  30'  52"  E.    Vel. = 50.253. 

The  process  is  of  course  abridged,  if  the  forces  act  at  a  right 
angle  with  each  other,  as  in  the  following  example : 

2.  A  balloon  rises  1120  feet  in  one  minute,  and  in  the  same 
time  is  borne  by  the  wind  370  feet;  what  angle  does  its  path  make 
with  the  vertical,  and  what  is  its  velocity  per  second  ? 

Ans.  18°  16' 53";  v  =  19.659. 


MECHANICS. 


In  the  next  example,  one  component  and  the  angle  which  each 
component  makes  with  the  resultant,  are  given  to  find  the  result- 
ant and  the  other  component. 

3.  From  an  island  in  the  Straits  of  Sunda,  we  sailed  S.  E.  by  S. 
(33°  45')  at  the  rate  of  6  miles  an  hour;  and  being  carried  by  a 
current,  which  was  running  toward  the  S.  "VV.  (making  an  angle 
with  the  meridian  of  64°  12  J'),  at  the  end  of  four  hours  we  came 
to  anchor  on  the  coast  of  Java,  and  found  the  said  island  bearing 
due  north ;  required  the  length  of  the  line  actually  described  by  the 
ship,  and  the  velocity  of  the  current  ? 

Ans.  s  =  26.4  miles. 

v  —  3.7024  miles  per  hour. 

If  the  magnitudes  and  directions  of  any  number  of  forces  are 
given,  the  resultant  of  them  all  is  obtained  by  a  repetition  of  the 
same  process  as  for  two.  In  Fig.  18,  first  calculate  A  C,  and  the 
angle  A  C  B,  by  means  of  A  B,  B  C9  and  the  angle  B.  Subtract- 
ing A  C  B  from  B  C  D,  we  have  the  same  data  in  the  next  tri- 
angle, to  calculate  A  D,  and  thus  proceed  to  the  final  resultant, 
AF. 

As  it  is  immaterial  in  what  order  the  components  are  intro- 
duced into  the  calculation,  it  will  diminish  labor,  to  find  first 
the  resultant  of  any  two  equal  compo- 
nents, or  any  two  which  make  a  right 
angle  with  each  other ;  since  it  can  be 
done  by  the  solution  of  an  isosceles,  or 
a  right-angled  triangle. 

4.  The  particle  A  (Fig.  20)  is  urged 
by  three  equal  forces  A  B,  A  0,  and 
A  D\   the  angle  BAG-  90°,  and 
GAD  —  45° ;  what  is  the  direction  of 
the  resultant,  and  how  many  times  A  B? 

Ans.  B  A  F=  80°  16',  and 
A  F  :  A  B  : :  4/8  :  1. 

5.  Five  sailors  raise  a  weight  by 
means  of  five  separate  ropes,  in  the  same 

plane,  connected  with  the  main  rope  that  is  fastened  to  the  weight 
in  the  manner  represented  in  Fig.  22.  B  pulls  at  an  angle  with 
A  of  20° ;  C  with  B,  19° ;  D  with  C,  21°  30' ;  and  E  with  D,  25°. 
A,  B,  and  (7,  pull  with  equal  forces,  and  D  and  E  with  forces  one- 
half  greater;  required  the  magnitude  and  direction  of  the  re- 
sultant. 

Ans.  Its  angle  with  A  is  46°  33'  10".  Its  magnitude  is  5.1957 
times  the  force  of  A. 

If  the  polygon  0  A  B  C  D  E  (Fig.  21)  be  constructed  for  the 
above  case,  A  G  and  D  F  are  easily  calculated  in  the  isosceles  tri- 


FIG.  20. 


CALCULATION    OF    A    COMPONENT. 


angles  ABC  and  D  E  F,  after  which  A  D  and  then  A  F  are  to 
be  obtained  by  the  general  theorem. 


FIG,  21. 


FIG.  22. 


FIG.  23. 


V 


46.  The  Resultant  and  all  Components,  except  one, 
being  given,  to  Find  that  one  Component.— If  A  B  (Fig. 
23)  is  the  resultant  to  be  produced,  and 
there  already  exists  the  force  A  C,  a 
second  force  can  be  found,  which  acting 
jointly  with  A  0,  will  produce  the  mo- 
tion required.  Join  C  B,  and  draw  A  D 
equal  and  parallel  to  it,  then  A  D  is  the 
force  required ;  for  A  B  is  equivalent  to 
A  <7and  C  B.  Therefore  C  B  has  the 
magnitude  and  direction  of  the  required 
force ;  A  D  is  the  line  in  which  it  must 
act. 

Again,  suppose  that  several  forces  act  on  A,  and  it  is  required 
to  find  the  force  which,  in  conjunction  with  them  all,  shall  pro- 
duce the,  resultant  A  B.  Let  the  several  forces  be  combined  into 
one  resultant,  and  let  A  C  represent  that  resultant.  Then  A  D 
may  be  found  as  before. 

The  trigonometrical  process  for  finding  a  component  is  essen- 
tially the  same  as  for  finding  a  resultant. 

1.  A  ferry-boat  crosses  a  river  |  of  a  mile  broad  in  45  minutes, 
the  current  running  all  the  way  at  the  rate  of  3  miles  an  hour ;  at 
what  angle  with  the  direct  course  must  the  boat  head  up  the  stream 
in  order  to  move  perpendicularly  across  ?  Ans.  71°  34'. 


30  MECHANICS. 

2.  A  sloop  is  bound  from  the  mainland  of  Africa  to  an  island 
bearing  W.  by  N.  (78°  45')  distant  76  miles,  a  current  setting 
K  N.  W.  (22°  30')  3  miles  an  hour;  what  is  the  course  to  arrive 
at  the  island  in  the  shortest  time,  supposing  the  sloop  to  sail  at  the 
rate  of  6  Sw8s^)er  hour ;  and  what  time  will  she  take ? 

Ans.  Course,  S.  76°  41'  4"  W.    Time,  10  h.  40m.  7  sec. 

3.  The  resultant  of  two  forces  is  10 ;  one  of  them  is  8,  and  the 
direction  of  the  other  is  inclined  to  the  resultant  at  an  angle  of  36°. 
Find  the  angle  between  the  two  forces. 

Ans.  47°  17'  5"  or  132°  42'  55". 

4.  A  ball  receives  two  impulses :  one  of  which  would  carry  it 
N.  27  feet  per  second ;  the  other,  E.  30°  N.  with  the  same  velocity ; 
what  third  impulse  must  be  conjoined  with  them,  to  make  the  ball 
go  E.  with  a  velocity  of  21  feet  ?     Ans.  S.  3°  22'  W.    v  =  40.57. 

47.  Resolution  of  Motion.— In  the  composition  of  motions 
or  forces,  the  resultant  of  any  given  components  is  found ;  in  the 
resolution  of  motion  or  force,  the  process  is  reversed;  the  resultant 
being  given,  the  components  are  found,  which  are  equivalent  to 
that  resultant. 

If  it  be  required  to  find  what  two  components  can  produce  the 
resultant  A  B  (Fig.  24),  we  have 
only  to  construct  on  A  B,  as  a  base, 
any  triangle  whatever,  as  A  B  C  or 
A  B  D  (Art.  40) ;  then,  if  A  G  is 
one  component,  the  other  is  A  F, 
equal  and  parallel  to  CB ;  or  if  A  D 
is  one,  the  other  is  A  E,  equal  and 
parallel  to  D  B ;  and  so  for  any  tri- 
angle whatever  on  the  base  A  B. 
The  number  of  pairs  is  therefore  in- 
finite, whose  resultant  in  each  case  is  A  B. 

The  directions  of  the  components  may  be  chosen  at  pleasure, 
provided  the  sum  of  the  angles  made  with  A  B  is  less  than  two 
right  angles. 

The  magnitude  and  direction  of  one  component  may  be  fixed 
at  pleasure. 

The  magnitudes  of  both  components  may  be  what  we  please, 
provided  their  difference  is  not  greater,  and  their  sum  not  less, 
than  the  given  resultant. 

These  conditions  are-obvious  from  the  properties  of  the  triangle. 

When  a  given  force  has  been  resolved  into  two  others,  each  of 
those  may  again  be  resolved  into  two,  each  of  those  into  two  others 
still,  and  so  on.  Hence  it  appears  that  a  given  force  may  be  re- 
solved into  any  number  of  components  whatever,  with  such  limi- 


FIG.  25. 


RESOLUTION    INTO    PAIRS.  31 

tations  as  to  direction  and  magnitude  as  accord  with  the  foregoing 
statements. 

1.  A  motion  of  153  toward  the  north  is  produced  by  the  forces 
100  and  125 ;  how  are  they  inclined  to  the  meridian  ? 

Am.  54°  28'  and  40°  37;  7". 

2.  A  resultant  of  617  divides  the  angle  between  its  components 
into  28°  and  74° ;  what  are  the  components  ? 

Ans.  606.34  and  296.14. 

48.  Resolution  into  Pairs,  with  Certain  Conditions. — 

In  some  cases,  in  which  a  condition  is  imposed,  a  simple  construe* 
tion  will  enable  us  to  find  the  pairs  of  components  which  fulfill 
that  condition. 

1st.  A  given  force  is  to  be  resolved  into  pairs  which  make  a 
given  angle  with  each  other. 

Let  A  B  (Fig.  25)  be  the  given  force.  On  A  B  as  a  chord, 
construct  the  segment  of  a  circle  A  D  B,  con- 
taining an  angle  equal  to  the  supplement  of 
the  given  angle.  Then  all  the  possible  pairs  of 
components  fulfilling  the  condition  will  be 
found  by  drawing  lines  from  A  and  B  to  points 
of  the  curve,  as  A  D,  D  B,  and  A  C,  C  B,  &c. 
The  segment  must  contain,  not  the  given  angle  itself,  but  its  sup- 
plement, because  the  given  angle  is  at  A,  between  A  D  and  a 
parallel  to  D  B,  or  G  B,  &c. 

If  the  given  angle  were  a  right  angle,  the  segment  to  be  con- 
structed is  a  semicircle. 

2d.  To  resolve  a  given  force  into  two  components,  making  a 
given  sum;  the  sum  must  not  be  less  than  the  given  force 
(Art.  47). 

Let  A  B  (Fig.  26)  be  the  given  force,  and  M  N  the  given 
sum.    Having  placed  A  B  on  M  N, 
so  that  A  N—  B  M,  construct  a  semi-  FlG-  26- 

ellipse  on  M  N  as  a  transverse  axis,  ^^- "xV\ 

with  A  and  B  for  the  foci.    Lines     D  .f  y^  Y>v 

drawn  from  A  and  B  to  any  point  of     /V^-^/  \     \ 

the  curve  will  represent  a  pair  of  the  /  ^^/^  *-^^\  \ 
required  components ;  for,  by  a  prop-  N  A  B  M 

erty  of  the  ellipse,  A  D  +  D  B,  or 
A  G  +  GB  =  M  N. 

3d.  To  resolve  a  given  force  into  two.  components,  having  a 
given  difference  ;  the  difference  must  not  be  greater  than  the  given 
force  (Art.  47). 

Let  A  B  (Fig.  27)  be  the  given  force,  and  M  N  the  given  dif- 
ference. Place  M  N  on  A  B  so  that  A  N  =  B  M,  and  construct 


32 


MECHANICS. 


the  hyperbola  M  C,  N  D,  having  M  Nfor  its  transverse  axis,  with 

A  and  B  for  foci.    Then  A  D,  D  B,  or  A  C,  G  B,  or  any  other 

lines  from  the  foci  to  a  point 

of  the  curve,  will  fulfill  the       D 

condition  required,  because 

their  difference  equals  the 

transverse  axis. 

It  is  required  to  resolve 
the  force  194  into  pairs  of 
components  acting  at  an  an- 
gle of  135°  with  each  other ; 
what  is  the  radius  of  the  circle  whose  segment  is  employed  in  the 
construction  ?  Let  A  B  =  194  (Fig.  28),  and  A  D  B  =  45° ; 
then  A  C  B  =  90°.  Let  G  H  be  perpendicular 

to  A  B\   then  A  B  :  A  C  : :  A  G  :  ^,  and 

T>, 
A     T> 

=  137.18. 


A 


2.  To  find  the  radius  of  the  circle  whose  seg- 
ment includes  the  components  of  the  force  a 
acting  at  any  given  angle  with  each  other. 
Make  A  B  —  a,  and  let  A  D  B  —  A,  the  sup- 
plement of  the  given  angle.  Then  A  C  £  =  2  A,an&A  Off 
therefore, 


—  A; 


sin  A  :  1  :  :  ~  :  A  G  = 


0         A  . 
2  sin  A 


49.  Resolution  of  a  Force,  to  Find  its  Efficiency  in  a 
Given  Direction. — By  the  resolution  of  a  force  into  two  others 
acting  at  right  angles  with  each  other,  it  is  ascertained  how  much 
efficiency  it  exerts  to  produce  motion  in  any  given  direction.  For 
example,  a  weight  W  (Fig.  29),  lying  on  a  horizontal  plane,  and 


pulled  by  the  oblique  force  C  A,  is  prevented  by  gravity  from 
moving  in  the  line  C  A,  and  is  compelled  to  remain  on  the  plane. 
Resolve  C  A  into  C  I>,  in  the  plane,  and  G  D  perpendicular  to  it ; 
then  the  former  represents  the  component  which  is  efficient  to 


RECTANGULAR    AXES 


33 


cause  motion  along  the  plane ;  the  latter  has  no  influence  to  aid  or 
hinder  that  motion ;  it  simply  diminishes  pressure  upon  the  plane. 
In  like  manner,  if  A  C  is  an  oblique  force,  pushing  the  weight,  its 
horizontal  component,  B  C,  is  alone  efficient  to  move  it;  the  other, 
A  B,  merely  increasing  the  pressure.  In  either  case,  the  whole 
force  is  to  that  component  which  is  efficient  to  move  the  body 
along  the  plane,  as  radius  to  the  cosine  of  inclination.  Also,  the 
whole  force  is  to  that  component  which  increases  or  diminishes 
pressure  on  the  plane,  as  radius  to  the  sine  of  inclination. 

If  only  88  per  cent,  of  the  strength  of  a  horse  is  efficient  in 
moving  a  boat  along  a  canal,  what  angle  does  the  rope  make  with 
the  line  of  the  tow-path?  Ans.  28°  21'  27". 

50.  Resultant  found  by  means  of  Rectangular  Axes.— 

When  several  forces  act  in  one  plane  upon  a  body,  their  resultant 
may  be  conveniently  found  by  the  use  of  right-angled  triangles 
alone.  Select  at  pleasure  two  lines  at  right  angles  to  each  other, 
both  of  them  lying  in  the  plane  of  the  forces,  and  passing  through 
the  point  at  which  the  forces  are  applied.  These  lines  are  called 
axes.  The  following  example  illustrates  their  use : 

Let  P  A,  P  B,  P  C,  P  D,  P  E  (Fig.  30)  represent  the  forces 
in  Question  5  (Art.  45).    Let  one  axis,  for 
convenience,  be  chosen  in  the  direction  P  A, 

and  let  P  H  be  drawn  at  right  angles  to  it       '  •  '  [  •' I^P 

for  the  other  axis.  These  axes  are  supposed 
to  be  of  indefinite  length.  Then  proceed  as 
in  Art.  49,  to  resolve  each  force  into  two 
components  on  these  axes.  As  P  A  acts  in 
the  direction  of  one  axis,  it  does  not  need  to 
be  resolved.  To  resolve  P  B,  say 

R  :  cos  20°  : :  P  B  :  P  b,  and 

P.- sin  20°  ::P  £  :  Pb': 


FIG.  80.' 

bed 


again, 


P:cos39°::P  (7:  PC,  and 
#:sin  39°::P  C:Pc',&c. 


Suppose  P  A  produced  so  as  to  equal  PA  +  Pb  +  Pc  + 
P  d  4-  P  e  =  M,  and  P  IT  produced  so  as  to  equal  P  b'  +  P  c'  + 
P  d'  +  P  e'  —  N.  Now,  as  M  acts  in  the  line  P  A,  and  N  at 
right  angles  to  it,  their  resultant  and  the  angle  which  it  makes 
with  P  A  are  found  by  the  solution  of  another  right-angled  tri- 
angle. The  resultant  is  5.1957,  and  the  angle  is  46°  33'  10",  as  in 
Art.  45. 

If  any  components  of  the  resolved  forces  are  opposite  to  P  A 
or  P  H,  they  are  reckoned  as  negative  quantities. 
3 


34 


MECHANICS. 


51.  Analytical  Expression  for  the  Resultant.-— Put  A  C 

(Fig.  31)  =  P,A  B  =  P',A  D  =  R,  angle  CA  B  =  a;  then  by 
Trig. 

A  Z>2  =  A  C*  +  CD*  -  2  A  C  x  CD  cos  A  CD,  or 
P2  =  P2  +  P'3  +  2  P  P'  cos  a;  whence 
R  =  VP*  +  P/2  +  2  P  P'  cos  a  . . .  (1).    Hence 
Tlie  resultant  of  any  two  forces,  act- 
ing at  the  same  point,  is  equal  to  the 
square  root  of  the  sum  of  the  squares 
of  the  two  forces,  plus  twice  the  pro- 
duct of  the  forces  into  the  cosine  of  the 
included  angle. 

If  a  =  0,  its  cosine  will  be  1,  and  -^ 
(1)  becomes 

•  R  =  P  +  P'. 
If  a  =  90°,  its  cosine  will  be  0,  and  we  shall  have 


FIG.  81. 


If  a  =  180°,  its  cosine  will  be  ~  1,  and  we  shall  have 
R  =  P  -  P'. 

1.  Two  forces,  P  and  P',  are  equal  in  intensity  to  24  and  30, 
respectively,  and  the  angle  between  them  is  105° ;  what  is  the  in- 
tensity of  their  resultant?  Ans.  33.21. 

2.  Two  forces,  P  and  P',  whose  intensities  are,  respectively, 
equal  to  5  and  12,  have  a  resultant  whose  intensity  is  13 ;  required 
the  angle  between  them.  Ans.  90°. 

3.  A  boat  is  impelled  by  the  current  at  the  rate  of  4  miles  per 
hour,  and  by  the  wind  at  the  rate  of  7  miles  per  hour ;  what  will 
be  her  rate  per  hour  when  the  direction  of  the  wind  makes  an 
angle  of  45°  with  that  of  the  current  ?  Ans.  10.2  miles. 

4.  Two  forces  and  their  resultant  are  all  equal;  what  is  the 
value  of  the  angle  between  the  two  forces  ?  Ans.  120°. 

52.  Principle  of  Moments. — The  moment  of  a  force,  with 
respect  to  a  point,  is  the 
product  of  the  force  into 
the  perpendicular  let  fall 
from  the  point  to  the 
line  of  direction  of  the 
force. 

The  fixed  point  is 
called  the  centre  of  mo- 
ments; the  perpendicu- 
lar distance,  the  lever- 
arm  of  the  force;  and 
the  moment  measures  the  tendency  of  the  force  to  produce  rotation 


FIG.  32. 


PARALLEL    FORCES.  35 

about  the  centre  of  moments.  Denote  the  two  forces  A  D,  A  B, 
and  their  resultant  A  0  (Fig.  32)  by  P,  P,  and  R,  respectively. 
From  E,  any  point  in  the  plane  of  the  forces,  let  fall,  upon  the 
directions  of  the  forces,  the  perpendiculars  E  F,  E  H,E  G.  Repre- 
sent these  perpendiculars  by  p,  p',  r.  Draw  D  K  and  B  L  perpen- 
dicular to  A  C.  Put  a=CAD,(3=CAB,0  =  CAE.  Then 

R  =  A  L  +  <7J&  =  P'cos0  +  Pcosa...(l); 
D  K  =  P'  sin  ft  B  L  =  P  sin  a;  .-.,  since  D  K  =  B  L, 
P'  sin  (3  =  P  sin  a,  or,  0  =  P'  sin  j3  —  P  sin  a . . .  (2). 
Multiplying  both  members  of  (1)  by  sin  0,  both  members  of  (2) 
by  cos  6,  adding  and  reducing,  we  have 

R  sin  6  =  P'  sin  (6  +  0)  +  P  sin  (6  -  a)  . . .  (3). 

But  sin  6  =  -j^,  sin   (0  +  0)  =  -^,  sin  (0  -  a)  =  -^; 

/.  (3)  reduces  to 

Er  =  P  p'  +  Pp...  (4). 

•  If  the  point  E  falls  within  the  angle  CAD,  sin  ((9  —  a)  be- 
comes negative,  and  (3)  becomes 

Rr  =  P'p'  -Pp...  (5). 

Hence,  the  moment  of  the  resultant  of  two  forces  is  equal  to  the 
algebraic  sum  of  the  moments  of  the  forces  taken  separately. 

53.     Forces    Acting    at    Different    Points.      Parallel 
Forces. — We  have  thus  far  considered  forces  acting  upon  a 
single  particle,  or  upon  one 
point  of  a  body.     If,  how- 
ever, two  forces  P  and  P', 
in  the  same  plane,  act  upon 
A    and    B,    two    different 
points  of  a  rigid  body,  they 
may  still  have  a  resultant. 

Let  the  lines  of  direc- 
tions of  the  two  forces  A  F  and  B  D  (Fig,  33)  be  produced  to 
meet  in  C.  The  two  forces  may  then  be  considered  as  acting  at 
C,  and  thus  compounded  into  a  single  force  at  that  point,  or  at 
the  point  G  of  the  body. 

By  (1)  of  the  last  article  this  resultant  is 

R  =  P'  cos  j3  4-  P  cos  a  .  . .  (1). 

"When  the  forces  become  parallel,  as  A  F  and  B  E,  (3  =  0,  and 
a  =  0,  and  (1)  becomes 

R  =  P'  +  P . . .  (2). 

If  the  parallel  forces  act  in  opposite  directions,  as  A  F  and 
B  E1,  then  a  =  180°,  and  (3  =  0,  and  (1)  becomes 
R  =  P'  -  P . .  .  (3).    Hence, 


36  MECHANICS. 

The  resultant  of  two  parallel  forces  is  in  a  direction  parallel  to 
them  and  equal  to  their  algebraic  sum. 

54.  Point  of  Application  of  the  Resultant.— Let  P  and 

P'  (Figs.  34,  35)  be  two  parallel  forces  acting  in  the  same  or  in 

FIG.  34  FIG.  35. 


opposite  directions,  and  let  E  be  the  point  of  application  of  the 
resultant.  Assume  this  point  as  a  centre  of  moments  ;  then  from 
(5)  of  Art.  52,  since  r  —  0, 

P  x  H  E  =  P'  x  G  E,  or,  in  the  form  of  a  proportion, 
P'  :  P  :  :  H  E  :  G  K    But  by  similar  triangles, 
HE-.  G  E\\A  E\E  B\  :. 
P  \P\\AE\E  B.     N 

That  is,  the  line  of  direction  of  the  resultant  of  two  parallel  forces 
divides  the  line  joining  the  points  of  application  of  the  components, 
inversely  as  the  components. 

By  composition  (Fig.  34)  and  division  (Fig.  35)  we  obtain  - 
P'  +  P  :  P  :  :  A  B  :  E  B,  and 
P  -  P'  :  P  :  :  A  B  :  E  B. 

That  is,  if  a  straight  line  ~be  drawn  to  meet  the  lines  of  two  parallel 
forces  and  their  resultant,  each  of  the  three  forces  will  be  propor- 
tional to  that  part  of  the  line  contained  betiveen  the  other  two. 
When  the  forces  act  in  the  same  direction,  we  have 

P  x  A  B 
E  B  =  -pj  -  p-,  and  when  they  act  in  opposite  directions, 


•   P-P" 

If,  in  the  last  case,  P  =  P',  then  E  B  will  be  infinite.  The 
two  forces  in  this  case  constitute  what  is  called  a  couple.  Their 
effect  is  to  produce  rotation  about  a  point  between  them. 

Any  number  of  parallel  forces  may  be  reduced  to  a  single  force 
(or  to  a  couple)  by  first  finding  the  resultant  of  two  forces,  then 
the  resultant  of  that  and  a  third  force,  and  so  on  to  the  last.  And 
any  single  force  may  be  resolved  into  two  or  any  number  of  paral- 
lel forces  by  a  method  the  reverse  of  this. 

55.  The  Parallelepiped  of  Forces.—  Hitherto  forces  have 
been  considered  as  acting  in  the  same  plane.  But  if  forces  act  in 


RECTANGULAR    AXES. 


37 


FIG. 


different  planes,  the  solution  of  every  case  may  be  reduced  to  the 
following  principle,  called  the  parallelepiped  of  forces. 

Any  three  forces  acting  in  different  planes  upon  a  body  may  be 
represented  by  the  adjacent  edges  of  a  parallelopiped,  and  their  re- 
sultant ~by  the  diagonal  which  passes  through  the  intersection  of 
those  edges. 

Let  A  C,AD,  and  A  E  (Fig.  36),  be  three  forces  applied  m 
different  planes  to  the  body  at  A. 
Construct  the  parallelepiped  C  P, 
having  A  C,  A  D,  and  A  E,  for  its 
adjacent  edges,  and  from  A  draw  the 
diagonal  A  B.  The  section  through 
the  opposite  edges  A  C  and  P  B  is  a 
parallelogram,  and  therefore  A  B  is 
the  resultant  of  A  C  and  A  P,  and 
A  P  is  the  resultant  of  A  D  and  A  E. 
Hence  A  B  is  the  resultant  of  A  C, 
A  D,  and  A  E. 

This  process  may  obviously  be  reversed,  and  a  given  force  may 
be  resolved  into  three  components  in  different  planes  along  the 
edges  of  a  parallelepiped,  having  such  inclinations  as  we  please. 

56.  Rectangular  Axes.— The  parallelepiped  generally 
chosen  is  that  whose  sides,  are  rectangles;  the  three  adjacent 
edges  of  such  a  solid  are  called  rectangular  axes.  All  the  forces 
which  can  possibly  act  on  a  body  may  be  resolved  into  equivalent 
forces  in  the  direction  of  three  such  axes.  And  since  all  forces 
which  act  in  the  direction  of  any  one  line  may  be  reduced  to  a 
single  force  by  taking  their  algebraic  sum,  therefore  any  number 
of  forces  acting  through  one  point  may  be  reduced  to  three  in  the 
direction  of  three  axes  chosen  at  pleasure. 

Let  A  X,  A  Y  (Fig.  37)  be  at  right  angles  with  each  other, 
FIG.  37.  FIG.  38. 


and  A  Z  perpendicular  to  the  plane  of  A  X  and  A  Y.    Let  A  B 
represent  a  force  acting  on  A.    Resolve  A  B  into  A  C  on  the  axis 


38 


MECHANICS. 


A  Z,  and  A  P  in  the  plane  of  A  X,  A  T\  then  resolve  A  P  into 
A  D  and  A  E  on  the  other  two  axes.  Therefore,  A  C,  A  D,  and 
A  E  are  three  rectangular  forces,  whose  resultant  is  A  B. 

Let  the  axes  A  X,  A  Y,  A  Z,  be  produced  indefinitely  (Fig.  38) 
to  X',  Y1,  Z';  then  their  planes  will  divide  the  angular  space 
about  A  into  eight  solid  right  angles,  namely  :  A-X  YZ,  A-X  Y'Z, 
A-X'  Y'Z,  A-X'YZ,  above  the  plane  of  X  and  Y}  and  A-X  Y  Z', 
A-X  Y'Z',  A-X'  Y'Z',  A-X'  Y  Z'  below  it. 

57.    Geometrical  Relation   of  Components   and    Re- 
sultant. —  A  force  acting  on  the  body 
A  may  be  situated  in  any  one  of  the 
eight  angles,  and  its  value  may  be 
expressed  in  terms  of  the  squares  of 
its  three  components.    Let  A  B  (Fig. 
39)  be  resolved  as  before  into  the 
rectangular  components  A  0,  A  D, 
and  A  E.     Then,  by  the  right-an- 
gled triangles,  we  find 
AB*=BP*+AP*=AE*+AP*; 
and 


FlG- 


.-.  A  B*  =  A  C*  +  A  D*  +  A 


and  A  B  =  VA  C*  +  A  D*  +  A  E\ 

If  A  B  is  in  the  plane  of  X  and  Y,  the  component  on  the  axis 
of  Z  becomes  zero,  and  A  B  =  VA  C*  +  A  Z>2,  and  similarly  for 
the  other  planes. 

58.  Trigonometrical  Relation  of  Components  and  Re- 
sultant. —  Let  the  angles  which  A  B  makes  with  the  axes  of 
X,  Y,  Z,  respectively,  be  a,  ft  y  ;  •  that  is,  B  A  C  =  a,  B  A  D  =  ft 
B  A  E  =  y.    In  the  triangle  ABC,  right-angled  at  (7,  we  have 
A  B  :  A  0  :  :  rad  :  cos  a  ;  therefore,  making  rad  =  1, 

A  C  =  A  B  .  cos  a. 

In  like  manner,  A  D  =  A  B  .  cos  (3  • 

and  A  E  —  A  B  .  cos  y. 

And  since  A  B  is  the  resultant  of  the  forces  A  C,  A  D,  and 
A  E,  it  is  the  resultant  of  A  B  .  cos  a,  A  B  .  cos  ft  A  B  .  cos  y. 
In  general,  the  components  of  any  force  P,  when  resolved  upon 
three  rectangular  axes,  are  P  .  cos  a,  P  .  cos  ft  P  .  cos  y. 

59.  Any  Number  of  Forces   Reduced   to  Three   on 
Three  Rectangular  Axes.  —  Suppose  the  body  at  A  to  be  acted 
upon  by  a  second  force  P',  whose  direction  makes  with  the  axes 
the  angles  a',  (3',  y'  ;  then,  as  before,  P'  is  the  resultant  of  P'.  cos  a', 
P'.cos  B',  P'.cos-}';  and  a  third  force  P",  in  like  manner,  has 


EQUILIBRIUM    OF    FORCES.  39 

for  its  components  F" .  cos  a",  F"  .  cos  A",  F"  .cos  y";  and  so  of 
any  number  of  forces. 

Now,  all  the  components  on  one  axis  may  be  reduced  to  one 
force  by  adding  them  together.  Hence,  the  whole  force  in  the 
axis  of  X  =  P .  cos  a  +  P1 .  cos  a'  +  P"  .  cos  a"  +  P'» .  cos  a'"  +  &c. ; 
the  whole  in  the  axis  of  Y, 

=  P .  cos  j3  +  P' .  cos  0'  +  P" .  cos  0"  +  P1" .  cos  /3'"  +  &c.; 
and  that  in  the  axis  of  Z, 

=  P .  cos  y  +  P' .  cos  y'  +  'P" .  cos  y"  +  P"' .  cos  y'"  -f  &c. 

If  any  component  acts  in  a  direction  opposite  to  others  in  the 
same  axis,  it  is  affected  by  a  contrary  sign,  so  that  the  force  in  the 
direction  of  any  axis  is  the  algebraic  sum  of  all  the  individual 
forces  in  that  axis. 

If  the  sum  of  the  components  in  one  axis  is  reduced  to  zero  by 
contrary  signs,  the  effect  of  all  the  forces  is  limited  to  the  plane  of 
the  other  axes,  and  is  to  be  obtained  as  in  Art.  50,  where  two  axes 
were  employed.  If  the  sum  of  the  components  on  each  of  two  axes 
is  reduced  to  zero,  then  the  whole  force  is  exerted  in  the  direction 
of  the  remaining  axis,  and  is  therefore  perpendicular  to  the  plane 
of  the  other  two. 

60.  Equilibrium  of  Forces. — 

1.  Two  forces  produce  equilibrium  when  they  are  equal  and  act 
in  opposite  directions. 

It  was  shown  (Art.  42)  that  two  forces  produce  the  least  re- 
sultant when  they  act  at  an  angle  of  180°  with  each  other,  and 
that  the  resultant  then  equals  the  difference  of  the  forces.  If  the 
forces  are  equal,  their  difference  is  zero,  and  the  resultant  vanishes ; 
that  is,  the  two  forces  produce  equilibrium. 

2.  Three  forces  produce  equilibrium  when  they  may  le  repre- 
sented in  direction  and  magnitude  ly  the  three  sides  of  a  triangle 
taken  in  order. 

For,  when  three  forces  are  in  equilibrium,  one  of  them  must 
be  equal  to,  and  opposite  to,  the  re- 
sultant  of   both   the  others.    But  FlG-  40- 
the  forces  A  G  and  A  B  (Fig.  40) 
produce  the  resultant  A  J9;  there- 
fore the  equal  and  opposite  force 
D  A,  since  it  is  in  equilibrium  with 
A  Z>,  is  also  in  equilibrium  with  A  C   -A  B 
and  A  B,Q?  A  C  and  C  D.    Hence 

the  three  forces  A  C,  C  D,  and  D  A,  taken  in  order  around  the 
figure,  produce  equilibrium. 


40  •  MECHANICS. 

It  is  obvious  that  three  forces  in  equilibrium  must  all  be  di- 
rected through  one  point,  else  each  force  could  not  be  opposed  to 
the  resultant  of  the  other  two. 

3.  More  than  three  forces  in  one  plane  will  produce  equilibrium 
when  they  can  be  represented  by  the  sides  of  a  polygon  taken  in 
order. 

In  Art.  43  it  was  shown  that  if  several  forces  acting  on  a  body, 
are  represented  by  all  the  sides  of  a  polygon  .except  one,  their  re- 
sultant is  represented  by  the  remaining  side.  Thus,  the  resultant 
of  the  forces  A  B,  B  C,  C  D,  and  D  E  (Fig.  41),  is  A  E.  Now, 
the  force  E  A,  equal  and  opposite  to  A  E,  since  it  would  be  in 
equilibrium  with  A  E,  is  therefore  in  equilibrium  with  all  the 
others.  Hence  the  forces  A  B,  B  C,  C  D,  D  E,  and  E  A,  taken 
in  order  around  the  figure,  are  in  equilibrium. 

FIG.  41.  FIG.  42. 


61.  Trigonometrical  Representation  of  Three  Forces 
in  Equilibrium. —  When  three  forces  are  in  equilibrium,  each  may 
be  represented  by  the  sine  of  the  angle  between  the  other  two. 

Let  A  E  (Fig.  42)  be  the  resultant  of  A  D  and  A  (7;  then,  if 
we  apply  a  force  A  B  equal  and  opposite  to  A  E,  the  forces  A  D, 
A  C,  and  A  B  will  be  in  equilibrium.  From  the  triangle  A  ED 
we  have  the  proportions 

A  D :  A  E :  E  D : :  sin  A  E  D :  sin  D  :  sin  E  A  D. 

But  sin  A  ED  =  sin  CA  E  =  sin  B  A  <7;  sin  D  —  sin  CA  D ; 
sin  E  A  D  =  sin  B  A  D;  A  E  =  A  B\  and  A  C  =  ED;  .-. 
AD'.AB-.A  CiismBA  tfrsin  CA  D-.smB AD. 

62.  Equilibrium  of  Parallel  Forces. — In  order  that  a  force 
may  be  in  equilibrium  with  two  parallel  forces, 

1.  It  must  be  parallel  to  them. 

2.  It  must  be  equal  to  their  algebraic  sum. 

3.  The  distances  of  its  line  of  action  from  the  lines  in  which  the 
two  forces  act,  must  be  inversely  as  the  forces. 

These  three  conditions  belong  to  the  resultant  of  two  parallel 
forces,  and  therefore  belong  to  that  force  which  is  in  equilibrium 
with  the  resultant. 


EQUILIBRIUM    OF    FORCES.  .41 

63.  Equilibrium  of  Couples. — If  two  parallel  forces  are 
such  as  to  constitute  a  couple,  no  one  force  can  be  in  equilibrium 
with  them.  For  the  resultant  of  a  couple  has  its  point  of  applica- 
tion at  an  infinite  distance  (Art.  54).  But  a  couple  can  be  held  in 
equilibrium  by  another  couple;  and  the  second  couple  maybe 
either  larger  or  smaller  than  the  given  couple,  or  it  may  be  equal 
to  it. 

Let  the  couple  P  and  P1  (Fig.  43)  act  FIG.  43. 

on  a  body  at  the  points  A  and  B ;  they  2 

tend  to  produce  rotation  about  the  middle 
point  C.    If  another  couple,  Q  and  Q', 

equal  to  P  and  P',  should  be  applied  to      

produce  equilibrium,  one  must  directly          P 
oppose  P,  and  the  other  P1.  Then  A  and 
B,  being  each  held  at  rest,  all  the  forces  JP-J, 

are  in  equilibrium. 

But  if  the  second  couple  is  less  than  B 

P  and  P',  they  must  act  at  distances  from 
(7,  which  are  as  much  greater  as  the  forces 

are  less ;  or,  if  the  second  couple  is  greater  ^ 

than  the  first,  they  must  act  at  distances  p> 

which  are  as  much  less.   Thus,  the  couple 

p  and  p'y  acting  at  D  and  E,  tend  to  produce  rotation  about  C  in 
one  direction,  and  P  and  P'  in  the  opposite ;  and  these  tendencies 
are  equal  when  D  C :  A  C :  :  P  :  p.  For,  since  the  opposite  forces, 
P  andjt?,  are  inversely  as  their  distances  from  (7,  their  resultant  is 
at  C,  and  is  equal  to  P  —  p  (Art.  53).  For  the  same  reason,  the 
resultant  of  P'  and  p'  is  at  C,  and  equal  to  P'  —  pr.  But  P  —  p  = 
P'  —  p',  and  they  act  in  opposite  directions.  Hence  C  is  at  rest, 
and  therefore  all  the  forces  are  in  equilibrium. 

64.  Equilibrium  of  Forces  in  Different  Planes. — Since 
all  the  forces  which  can  operate  on  a  body  may  be  reduced  to  three 
forces  on  rectangular  axes,  it  is  obvious  that  the  whole  system  of 
forces  cannot  be  in  equilibrium  till  the  sum  of  the  components  on 
each  axis  is  reduced  to  zero.    "We  must  have,  therefore,  in  Art.  59, 
as  conditions  of  equilibrium,  these  three  equations  for  the  three 
axes,  X,  ¥,  and  Z: 

P. cos  a  +  P' .  cos  a'  +  P" .  cos  a"  +,  &c.,  =  0 ; 

P.cos0+  P'.cosjG'  +  P" .  cos/5"  +,&c.,  =  0; 

^      P. cosy  +  P'.cos/-+  P".cos7"  +,&c.,  =  0. 

65.  Forces  Resisted  by  a  Smooth  Surface.— Whenever 
any  forces  cause  pressure  upon  a  smooth  surface,  and  are  held  in 
equilibrium  by  its  resistance,  the  resultant  of  those  forces  must  be 


MECHANICS. 


at  right  angles  to  the  surface.  Suppose  that  D  A  (Fig.  44)  is 
either  a  single  force  or  the  resultant  of 
two  or  more  forces,  and  that  it  is  held 
in  equilibrium  by  the  reaction  of  A  B, 
a  smooth  surface.  If  D  A  is  not  perpen- 
dicular to  the  surface,  it  can  be  resolved 
into  two  components,  one  perpendicular 
to  the  surface  A  B9  the  other  parallel  to 
it.  The  former,  D  B,  is  neutralized  by 
the  resistance  of  the  surface ;  the  latter,  B  A,  is  not  resisted,  and 
produces  motion  parallel  to  the  surface,  contrary  to  the  supposi- 
tion. 'Therefore  D  A,  if  held  in  equilibrium  by  the  surface  A  B, 
must  be  perpendicular  to  it. 


CHAPTER   IV. 

THE    CENTRE    OF    GRAVITY. 

66.  The  Centre  of  Gravity  Defined.— In  every  body  and 
in  every  system  of  bodies,  there  is  a  point  so  situated  that  all  the 
parts  acted  on  by  the  force  of  gravity  balance  each  other  about  it 
in  every  position.    That  point  is  called  the  centre  of  gravity.    The 
force  of  gravity  acts  in  parallel  lines  on  every  particle  of  a  body ; 
the  centre  of  gravity  must  therefore  be  the  point  through  which 
the  resultant  of  all  these  parallel  forces  is  directed,  in  every  posi- 
tion of  the  body.     Hence,  if  the  centre  of  gravity  is  supported,  the 
body  is  supported.    As  to  the  support  of  the  body,  therefore,  we 
may  imagine  all  parts  of  it  to  be  collected  in  its  centre  of  gravity. 
When  a  system  of  bodies  is  considered,  they  are  conceived  to  be 
united  to  each  other  by  inflexible  rods,  which  are  without  weight. 

67.  Centre  of  Gravity  of  Equal  Bodies  in  a  Straight 
Line. — The  centre  of  gravity  of  two  equal  particles  is  in  the 
middle  point  between  them.     Let  A  and  B  (Fig.  45),  two  equal 
particles,  be  joined  by  a  straight  line,  and  let 

A  a  and  B  b  represent  the  forces  of  gravity. 

The  resultant  of  these  forces,  since  they  are 

parallel  and  equal,  will  pass  through  the  mid-   a- 

die  of  A  B  (Art.  54) ;  G  is  therefore  the  centre 

of  gravity.     In  like  manner  it  is  proved  that 

the  centre  of  gravity  of  two  equal  bodies  is 

in  the  middle  point  between  their  respective  centres  of  gravity. 


FIG.  45. 

Or 


CENTRE    OF    GRAVITY.  43 

Any  number  of  equal  particles  .or  bodies,  arranged  at  equal 
distances  on  a  straight  line,  have  their  common  centre  of  gravity 
in  the  middle;  since  the  above  reasoning  applies  to  each  pair, 
taken  at  equal  distances  from  the  extremes.  Hence,  the  centre  of 
gravity  of  a  material  straight  line  (e.g.,  a  fine  straight  wire)  is  in 
the  middle  point  of  its  length. 

68.  Centre  of  Gravity  of  Regular  Figures.— In  the  dis- 
cussion of  the  centre  of  gravity  in  relation  inform,  bodies  are  con- 
sidered uniformly  dense,  and  surfaces  are  regarded  as  thin  lamina 
of  matter. 

In  plane  figures  the  centre  of  gravity  coincides  with  the  centre 
of  magnitude,  when  they  have  such  a  degree  of  regularity  thai  there 
are  two  diameters,  each  of  which  divides  the  figure  into  equal  and 
symmetrical  parts. 

The  circle,  the  parallelogram,  the  regular  polygon,  and  the 
ellipse,  are  examples. 

For  instance,  the  regular  hexagon  (Fig.  46)  is  divided  sym- 
metrically by  A  B,  and  also  by  CD.  Conceive 
the  figure  to  be  composed  of  material  lines 
parallel  to  A  B.  Each  of  these  has  its  centre- 
of  gravity  in  its  middle  point,  that  is,  in  C  D, 
which  bisects  them  all  (Art.  67).  Hence,  the 
centre  of  gravity  of  the  whole  figure  is  in  C  D. 
For  the  same  reason  it  is  in  A  B.  It  is,  there- 
fore, at  their  intersection,  which  is  also  the 
centre  of  magnitude. 

By  a  similar  course  of  reasoning  it  is  shown  that  in  solids  of 
uniform  density,  which  are  so  far  regular  that  they  can  be  divided 
symmetrically  by  three  different  planes,  the  centres  of  gravity  and 
magnitude  coincide ;  e.g.,  the  sphere,  the  parallelopiped,  the  cylin- 
der, the  regular  prism,  and  the  regular  polyhedron. 

69.  Centre  of  Gravity  between  Two  Unequal  Bodies. — 

The  centre  of  gravity  of  two  unequal  bodies  is  in  a  straight  line 

joining  their  respective  centres  of  gravity,  and  at  the  point  which 

divides  their  distance  in  the  inverse  ratio  of  their  weights.    Let 

A  a  and  B  b  (Fig.  47),  passing  through  the      A      _. 

centres  of  gravity  of  A  and  B,  be  proportional 

to  their  weights,  and  therefore  represent  the 

forces  of  gravity  exerted  upon  them.    By  the 

laws  of  parallel  forces,  the  resultant  G  g  = 

A  a  +  B  I  (Art.  53),  and  A  a  :  B  I : :  B  G  : 

A  G.  Therefore  the  centre  of  gravity  must  be 

at  G,  through  which  the  resultant  passes  (Art. 

66).    This  obviously  includes  the  case  of  equal  weights  (Art  67). 


44  MECHANICS. 

It  appears  from  the  foregoing  that  the  whole  pressure  on  a 
support  at  G  is  A  +  B,  and  that  the  system  is  kept  in  equilibrium 
by  such  support. 

70.  Equal  Moments  with  Respect  to  the  Centre  of 
Gravity.— If  A  is  put  for  the  weight  of  A,  and  B  for  that  of  B9 
the  above  proportion  becomes  A  :  B  : :  B  G  :  A  G. 

Let  the  proportion  be  changed  to  an  equation,  and  we  have 
A  xAG  =  BxBG.  Suppose  now  that  A  B  is  an  inflexible 
rod,  without  weight,  and  free  to  revolve  about  G.  Since  the 
bodies  balance  each  other  about  that  point,  the  equal  products, 
A  x  A  G  and  B  x  B  G,  may  be  taken  to  represent  the  equal  ten- 
dencies of  the  bodies  to  turn  the  system  about  G.  The  tendency 
of  the  body  A,  expressed  by  A  x  A  G,  is  called  the  moment  of  A, 
with  reference  to  the  point  G',  and,  similarly,  B  x  B  G  is  the  mo- 
ment of  B,  with  reference  to  the  same  point.  Hence  the  proposi- 
tion, that  the  moments  of  two  bodies  with  reference  to  their  centre 
of  gravity  are  equal. 

71.  Centre    of    Gravity   between    Three    or    More 
Bodies. — The  method  of  determining  the  centre  of  gravity  of 
two  bodies  may  be -extended  to  any  number. 

Let  A,  B,  C,  D,  &c.  (Fig.  48),  be  the  weights  of  the  bodies,  and 
let  the  centres  of  gravity  of  A  and  B  be  con- 
nected together  by  the  inflexible  line  A  B. 

Divide  A  B  so  that  A  :  B  : :  B  G  :  A  G,  or 
A  +  B:  B'.-.AB:  AG',  then  G  is  the  centre 
of  gravity  of  A  and  B.  Join  C  G ;  and  since 
A  +  B  may  be  considered  as  at  the  point  G,  *c  D 

divide  C  G  so  that  A  +  B  -f  C :  C :  :  C  G  :  G  g.  In  like  manner, 
K,  the  centre  of  gravity  of  four  bodies,  is  found  by  the  proportion, 
A  +  B  -f  C  -f  D\  D  : :  D  g  :  g  K.  The  same  plan  may  be  pur- 
sued for  any  number  of  bodies. 

72.  Centre  of  Gravity  of  a  Triangle. — The  centre  of  gravity 
of  a  triangle  is  one-third  of  the  distance  from  the  middle  of  a  side 
to  the  opposite  angle.    Bisect  A  C  in  D  (Fig.  49),  and  B  C  in  E\ 
join  A  E,  B  D,  and  D  E.     B  D  bisects  all  lines 

across  the  triangle  parallel  to  A  G\  therefore  the 
centre  of  gravity  of  all  those  lines — that  is,  of  the 
triangle — is  in  B  D.  For  a  like  reason,  it  is  in 
A  E,  and  therefore  at  their  intersection,  G.  Since 

But*  E  G  D  and  A  G  B  are  similar;  .-.  D  G  : 
BG::DE:AB:\  1  :  2;  .\D  G  =  A  B  G  =  |  BD. 

73.  Centre  of  Gravity  of  an  Irregular  Polygon.— Divide 

the  polygon  into  triangles  by  diagonals  drawn  through  one  of  its 


CENTRE    OF    GRAVITY. 


45 


angles,  and  then  proceed  according  to  the  methods  already  given. 
Let  A  C E  (Fig.  50)  be  an  irregular  polygon,  whose  centre  of  grav- 
ity is  to  be  found.  Divide 
it  into  the  triangles  P,  Q, 
R,  S9  by  diagonals  through 
A,  and  find  their  centres  of 
gravity  a,  b,  c,  d  (Art.  72). 
Join  a  ft,  and  divide  it  so 
that  a  b\a  G::P+Q:Q-, 
then  G  is  the  centre  of 
gravity  of  the  quadrilateral 
P  +  Q.  Then  join  G  c,  and 
make  Gc:  G  g  \  :P+Q+R 
:  R.  By  proceeding  in  this 
manner  till  all  the  triangles 

are  used,  the  centre  of  gravity  of  the  polygon  is  found  at  the  last 
point  of  division. 

74.  Centre  of  Gravity  of  the  Perimeter  of  an  Irregu- 
lar Polygon.— Find  the  centre  of  gravity  of  each  side,  which  is 
at  its  middle  point,  and  then 

proceed  as  in  Art.  71,  the 

weight  of  each  line  being 

considered  proportional  to 

its  length.    Thus,  let  a,  by 

c,  &c.,   be  the  centres  of 

gravity  of  the  sides,  A  B, 

EC,  CD,  &c.  (Fig.  51) ; 

join  a  b,  and  divide  it  so 

that  a*:  a  G:\AB  +  B  G 

\B  C\  then  G  is  the  centre 

of  gravity  of  A  B  and  B  C. 

Xext  join  G  c,  and  make  G  c 

then  g  is  the  centre  of  gravity  of  those  three  sides. 

this  manner  till  all  the  sides  are  used. 

The  perimeter  of  a  polygon  having  the  degree  of  regularity  de- 
scribed in  Art.  68,  has  its  centre  of  gravity  at  the  centre  of  the 
figure,  as  may  be  easily  proved.  If  a  polygon  has  a  less  degree  of 
regularity  than  that,  the  centre  of  gravity  both  of  its  area  and  its 
perimeter  may  usually  be  found  by  methods  more  direct  and 
simple  than  those  given  for  polygons  wholly  irregular. 

75.  Centre   of  Gravity  of  a  Pyramid.— The  centre  of 
gravity  of  a  triangular  pyramid  is  in  the  line  joining  the  vertex 
and  the  centre  of  gravity  of  the  base,  at  one-fourth  of  the  distance 
from  the  base  to  the  vertex. 


AB  +  BC  +  CD 

Proceed  in 


46 


MECHANICS. 


Let  G  (Fig.  52)  be  the  centre  of  gravity  of  the  base  B  D  C\  and 
f/  that  of  the  face  ABC.  The  line  A  G-  passes  through  the  cennv 
of  gravity  of  every  lamina 
parallel  to  D  B  (7,  on  account 
of  the  similarity  and  similar 
position  of  all  those  laminae ; 
/.  the  centre  of  gravity  of  the 
pyramid  is  in  A  G.  For  a 
similar  reason,  it  is  in  D  g ; 
and  therefore  at  their  inter- 
section, 0.  TSwEG^lE 
and  Eg  =  J  E  A ;  hence,  by 


similar  triangles,  #  G  =  \  A  D. 
But  Gg  0  and  AOD  are  also 
similar; .-.  G  0=-J  A  0=-\AG. 

From  this  it  is  readily  proved  that  the  centre  of  gravity  of  every 
pyramid  and  cone  is  one-fourth  of  the  distance  from  the  centre  of 
gravity  of  the  base  to  the  vertex. 

76.  Examples  on  the  Centre  of  Gravity. — 

1.  A,  B,  and   G  (Fig.  53),  weigh,  respectively,  3,  2,  and  1 
pounds,  A  B  =  5  ft,  B  C  =  4  ft.,  and 

C  A  =  2  ft.    Find   the  distance  of 
their  centre  of  gravity  from  C. 

First,  from  the  given  sides  of  the 
triangle  A  B  C,  calculate  the  angles. 
A  is  found  to  be  49°  274'.  Next  find 
the  place  of  G,  the  centre  of  gravity  of 
A  and  B,  bythe  proportion,  A  +  B  :  B  : :  A  B  :  A  6;  A  G  is  2 
feet,  equal  to  A  C.  Calculate  C  G,  the  base  of  the  isosceles  tri- 
angle A  G  C.  Its  length  is  1.673.  Then  find  Cg  by  the  propor- 
tion  C  G  :  Cg  : :  A  +  B  +  C:  A  +  B\  therefore  Cg  =  1.394. 

2.  A  =  5  Ibs.,  B  =  3  Ibs.,  and  C  =  12 Ibs. ;  A  £  =  8  ft,AO  = 
4  ft.,  and  the  angle  A  is  90° ;  find  the  distance  of  the  centre  of 
gravity  of  A,  B,  and  C,  from  C.  Ans.  2  ft. 

3.  Three  equal  bodies  are  placed  at  the  angles  of  any  triangle 
whatever ;  show  that  the  common  centre  of  gravity  of  those  bodies 
coincides  with  the  centre  of  gravity  of  the  triangle. 

4.  Find  the  centre  of  gravity  of  five  equal  heavy  particles 
placed  at  five  of  the  angular  points  of  a  regular  hexagon. 

Ans.  It  is  one-fifth  of  the  distance  from  the  centre  to  the 
third  particle. 

5.  A  regular  hexagon  is  bisected  by  a  line  joining  two  opposite 
angles;  where  is  the  centre  of  gravity  of  one-half? 

Ans.  Four-ninths  of  the  distance  from  the  centre  to  the 
middle  of  the  second  side. 


CENTRE    OF    GRAVITY.  47 

6.  A  square  is  divided  by  its  diagonals  into  four  equal  parts,  one 
of  which  is  removed ;  find  the  distance  from  the  opposite  side  of 
the  square  to  the  centre  of  gravity  of  the  remaining  figure. 

Ans.  T7S  of  the  side  .of  the  square. 

7.  Two  isosceles  triangles  are  constructed  on  opposite  sides  of 
the  same  base,  the  altitude  of  the  greater  being  h,  and  of  the  less, 
h' ;  where  is  the  centre  of  gravity  of  the  whole  figure  ? 

Ans.  On  the  altitude  of  the  greater  triangle,  at  a  distance 
from  the  cpmnion  base  equal  to  |  (li  —  h'). 

8.  The  base  and  the  place  of  the  centre  of  gravity  of  a  triangle 
being  given,  required  to  construct  the  triangle. 

9.  Given  the  base  and  altitude  of  a  triangle ;  required  to  con- 
struct the  triangle,  when  its  centre  of  gravity  is  perpendicularly 
over  one  end  of  the  base. 

10.  On  a  cubical  block  stands  a  square  pyramid,  whose  base, 
volume,  and  mass  are  respectively  equal  to  those  of  the  cube ; 
where  is  the  centre  of  gravity  of  the  figure  ? 

Ans.  One-eighth   of  the  height  of  the  cube  above  its 
upper  surface. 

77.  Centre  of  Gravity  of  Bodies  in  a  Straight  Line 
referred  to  a  Point  in  that  Line. — If  several  bodies  are  in  a 
straight  line,  their  common  centre  of  gravity  may  be  referred  to  a 
point  in  that  line ;  and  its  distance  from  that  point  is  obtained  by 
multiplying  each  iveight  into  its  oivn  distance  from  the  same  point, 
and  dividing  tlie  sum  of  the  products  by  the  sum  of  the  weights. 
Let  Af  B,  (7,  and  D,  represent  the  weights  of  several  bodies/ whose 
centres  of  gravity  are  in  the  straight  line  0  D  (Fig.  54).  Eequired 

FIG.  54. 
o  A  B  c  r> 

-r • 


the  distance  of  their  common  centre  of  gravity  from  any  point  0 
assumed  in  the  same  line.  Let  G  be  their  common  centre  of 
gravity,  then  the  moments  of  A  and  B  must  be  equal  to  the  op- 
posing moments  of  C  and  D  with  reference  to  the  point  G  (Art.  70). 
That  is, 

AxAG  +  Bx£G  =  CxCG  +  DxDG-,  or, 

A  x  (OG-OA)  +  B  x  (OG-0£)=  C  x  (OC-0  G) +D x 
(OD-  00)', 

expanding,  transposing  the  negative  products,  and  factoring,  we 
have 

O  G=AxOA+BxOB+  Cx  OC+DxOD. 


48 


MECHANICS. 


Therefore, 


00  = 


AxOA+BxOB+CxOC+DxOD 

A  +  B  + C+D 


78.  Centre  of  Gravity  of  a  System  referred  to  a 
Plane. — If  the  bodies  are  not  in  a  straight  line,  they  may  be  re- 
ferred to  a  plane,  which  is  assumed  at  pleasure.  The  distance  of 
their  common  centre  of  gravity  from  that  plane  is  expressed  as 
before :  multiply  each  weight  into  its  own  distance  from  the  plane, 
and  divide  the  sum  of  the  products  ~by  the  sum  of  the  bodies. 

Letp,p',p"  (Fig.  55),  represent  the  weights  of  several  bodies, 
whose  centres  of  gravity  are  at  those  points  respectively,  and  let 
A  C  be  the  plane  of  reference.  Join  p  p',  and  let  g  be  the  com- 
mon centre  of  gravity  of  p  and 
p' ;  draw  p  x,  g  k,  p'  x  at  right 
angles  to  the  plane  A  C,  and 
consequently  parallel  to  each 
other ;  join  x  x',  and  since  the 
points  p,  g,  p',  are  in  a  straight 
line,  the  points  x,  k,  x'  will  also 


FIG.  55. 


p 


be  in  a  straight  line,  and  there- 
fore $  x'  will  pass  through  &. 
Join  g  p")  and  let  0  be  the  com- 
mon centre  of  gravity  ofp,p',p"  ; 
draw  G  K,p"  x",  perpendicular 
to  the  plane  ;  and  through  g 
draw  m  n  parallel  to  x  x1  meet- 
ing p  x  produced  in  n. 


\ 


K 


sim.  triangles)  p'  m  :  p  n  ; 
x  p'  m,orp 


p 


x  (n  x  —  p  x)  =  p'  x  (p1  x'  —  m  x')  ; 
(g  k  —  p  x)  =  pf  x  (pr  x'  —g  &), 


.:p  x  p  n  =  p 
but 

n  x  =  g  Jc  = 
and 

,v          ,  ,       p  x  n  x  +  ft'  x  p'  x' 

(p+p)  xgk=pxpx  +  p'  x  p'  x'  .-.  g  Jc  =  ^    2      +J        —  ; 

for  the  same  reason,  if  p  +  p'  is  placed  at  g,  we  have 

(p+p')xgk+p"xp"x"  __pxp  x+p'xp'x'+p"xp"  x" 
'" 


p+p' 
a  formula  which  is  applicable  to  any  number  of  bodies. 

Let  the  last  equation  be  multiplied  by  the  denominator  of  the 
fraction,  and  we  have 

(p+p'+p"  +  &c.)  G  K=pxp  x+p'xp'  x'+p"xp"  a"  +  &c.; 
that  is,  the  moment  of  any  system  of  bodies  with  reference  to  a  given 


CENTRE    OF    GRAVITY.  49 

plane,  equals  the  sum  of  the  moments  of  all  the  parts  of  the  system 
with  reference  to  the  same  plane. 

79.  Centre  of  Gravity  of  a  Trapezoid,  —  As  an  example 
of  the  foregoing  principle,  let  it  be  proposed  to  find  the  centre  of 
gravity  of  a  trapezoid,  considered 
as  composed  of  two  triangles.  The 
centre  of  gravity  of  the  trapezoid 
A  C  (Fig.  56)  is  in  E  F,  which  bi- 
sects all  the  lines  of  the  figure 
parallel  to  B  C.  Suppose  G  to  be 
the  centre  of  gravity  of  the  trape- 
zoid ;  through  G  draw  K  M  per- 
pendicular to  the  bases.  Let  KM— 
h,  B  C  =£,  A  D  =  b,  and  join  B  D. 

The  moment  of   the  trapezoid  with  reference    to  B  G   is 

(B  +  1}  \  .  G  K    The  moment  of  the  upper  triangle  is  ~  .  f  />  ; 

*>  /v        O 

the  moment  of  the  lower  triangle  is  -^  -  .  ^  ;  /.  (B  +  b)  ~  .  G  K  — 

AI  O  fy 

B  h    h       I  h    2  .       , 

T  .  3+^.3  A;  whence 

B  +  2b    h  -    -  B  +  2  &    h 


±f  +  0        o 

By  similar  triangles 

GM-.GK'.iEG-.GF;  .:£  G  :  G  F::2  B  +  b:  E  +  2b;  or 
the  centre  of  gravity  of  a  trapezoid  is  on  tfie  line  which  bisects  the 
parallel  bases,  and  divides  it  in  the  ratio  of  twice  the  longer  plus  the 
shorter  to  twice  the  shorter  plus  the  longer. 

1.  Four  bodies,  A,  B,  C,  D,  weighing,  respectively,  2,  3,  6,  and 
8  pounds,  are  placed  with  their  centres  of  gravity  in  a  right  line, 
at  the  distance  of  3,  5,  7,  and  9  feet  from  a  given  point  ;  what  is 
the  distance  of  their  common  centre  of  gravity  from  that  given 
point  ;  and  between  which  two  of  the  bodies  does  it  lie  ? 

Ans.  Between  C  and  D  ;  and  its  distance  from  the  given 
point  7-j33  feet. 

2.  There  are  five  bodies,  weighing,  respectively,  1,  14,  2U,  22, 
and  29£  pounds  ;  a  plane  is  assumed  passing  through  the  last 
body,  and  the  distances  of  the  other  four  from  the  plane  are,  re- 
spectively, 21,  5,  6,  and  10  feet;  how  far  from  the  plane  is  the 
common  centre  of  gravity  of  the  five  bodies  ?  Ans.  5  feet. 

[See  Appendix  for  calculations  of  the  place  of  the  centre  of 
gravity  of  curvilinear  bodies.] 
4 


50  MECHANICS. 

80.  Centrobaric  Mensuration. — The  properties  of  the  centre 
of  gravity  furnish  a  very  simple  method  of  measuring  surfaces  and 
solids  of  revolution.  This  method  is  comprehended  in  the  two 
following  propositions,  known  as  the  theorems  of  dnldi-nuo; 

1.  If  any  line  revolve  about  a  fixed  axis,  which  is  in  the  plane  of 
that  line,  the  SURFACE  which  it  generates  is  equal  to  the  product 
of  the  given  line  into  the  circumference  described  by  its  centre  of 
gravity. 

Let  any  line,  either  straight  or  curved,  revolve  about  a  fixed 
axis  which  is  in  the  plane  of  that  line;  and  let  /,/',/",/";,  etc., 
denote  elementary  portions  of  the  line,  d,  d',  d",  d'",  &c.,  the  dis- 
tances of  these  portions,  respectively,  from  the  axis;  then  the 
surface  generated  by  /,  in  one  revolution,  will  be  2  TT  df;  hence 
the  surface  generated  by  the  whole  line  will  be 

S=%K(df+  d'f  +  d"f"  +  d'"f"  +  &c.)  .  . .  (1). 
Put  L  =  the  length  of  the  revolving  line,  and  G  =  the  dis- 
tance from  the  axis  to  the  centre  of  gravity  of  the  line ;  then 
(Art.  78) 

GL  =  df  +  d'f  +  d"f"  +  ft7"'/'"  +  &c (2). 

Combining  (1)  and  (2),  we  have 

S=27rGL...  (3). 

2.  If  a  plane  surface,  of  any  form  whatever,  revolve  about  a  fixed 
axis  which  is  in  its  own  plane,  the  VOLUME  generated  is  equal  to 
the  product  of  that  surface  into  the  circumference  described  by  its 
centre  of  gravity. 

Let  any  plane  surface  revolve  about  an  axis  which  is  in  the 
plane  of  that  surface;  and  let/,/',/17,/'7',  &c.,  denote  elementary 
portions  of  the  surface,  d,  d',  d",  d1",  &c.,  the  distances  of  these 
portions,  respectively,  from  the  axis ;  then  the  volume  generated 
by  /in  one  revolution  will  be  2  TT  df',  hence  the  volume  generated 
by  the  whole  surface  will  be 

V  =  2  TT  (df  +  d'f  +  d"f"  -f  d'"f"  +  &c.)  . .  .  (4). 

Fut  A  —  the  area  of  the  revolving  surface,  and  G  =  the  dis- 
tance from  the  axis  to  the  centre  of  gravity  of  that  surface ;  then 
(Art  78) 

AG  =  df  +  d'f  +  d"f"  +  d7"/ '"  +  &c., . . .  (5). 

Substituting  in  (4),  we  have 

V=%7rA  0...(6). 

As  an  illustration  of  the  first  theorem,  the  straight  line  C  D 
(Fig.  57),  revolving  about  the  centre  C,  describes  a  circle  whose 
surface  is  equal  to  C  D  into  the  circumference  of  the  circle  de- 
scribed by  its  centre  of  gravity,  E,  This  is  evident  also  from  the 


CENTROBARIC    MENSURATION.  51 

consideration  that,  since  E  is  the  centre  of  the  line  C  D,  the  cir- 
cumference described  by  it  will  -be  half  the 
length  of  the  circumference  A  D  B ;  and  the  FIG.  57. 

area  of  a  circle  is  equal  to  the  product  of  the 
radius  into  half  the  circumference. 

The  second  theorem  is  illustrated  by  the 
volume  of  a  cylinder,  whose  height  —  Ji,  and 
the  radius  of  whose  base  =  r. 

Common  method ;  base  =  TT  r* ;  height=A ; 
/.  yol.  =  TT  ra  h. 

Centrobaric  method ;  revolving  area  =  r  Ji ;  circumference  de- 
scribed by  the  centre  of  gravity  —  -J-  r  x  2  n ;  /.  vol.  =  r  h  .  £  r  . 
2n  =  TTT*  h. 

81.  Examples. — 

1.  Suppose  the  small  circle  (Fig.  57)  to  be  placed  with  its 
plane  perpendicular  to  the  plane  of  the  paper,  and  revolved  about 
(7,  the  point  D  describing  the  line  D  B  A  ;  required  the  content 
of  the  solid  ring.    If  C  D  —  R,  and  E  D  —  r,  then  the  area  re- 
volved —  TT  r2,  and  the  circumference  DBA=2rrR-)  .-.  the  ring 
=  2  TT2  R  r2.    It  is  equal  to  a  cylinder  whose  base  is  the  circle 
E  Z>,  and  whose  height  equals  the  line  DBA. 

2.  Find  the  convex  surface  of  a  cone ;  slant  height  =  s ;  and 
rad.  of  base  =  r.    The  line  revolved  being  s,  and  the  distance 
from  the  axis  to  its  centre  of  gravity,  ^  r,  the  surface  is  re  r  s. 

3.  A  square,  whose  side  is  one  foot,  is  revolved  about  an  axis 
which  passes  through  one  of  its  angles,  and  is  parallel  to  a  diago- 
nal ;  required  the  volume  of  the  figure  thus  formed. 

Ans.  TT  V2,  or  4.4429  cubic  ft. 

4.  Find  the  surface  of  a  sphere  whose  radius  is  r.     (The  dis- 
tance from  the  centre  of  a  circle  to  the  centre  of  gravity  'of  its 

2  r 
semi-circumference  is  — .    Appendix,  Art.  92.) 

2r    _ 

Ans.  TT  r  .  —  .  2  ff  =  4  TT  r . 

7T 

5.  Find  the  volume  of  a  sphere  whose  radius  is  r.    (Appendix, 
Art.  95.)  * ,  Ans.  |  TT  r3. 

82.  Support  of  a  Body. — A  body  cannot  rest  on  a  smooth 
plane,  unless  it  is  horizontal;  for  the  pressure  on  a  plane  (Art.  65) 
cannot  be  balanced  by  the  resistance  of  that  plane,  except  when 
perpendicular  to  it ;  therefore,  as  the  force  of  gravity  is  vertical, 
th'>  resisting  plane  must  be  horizontal. 

The  base  of  support  is  that  area  on  the  horizontal  plane  which 
is  comprehended  by  lines  joining  the  extreme  points  of  contact. 


MECHANICS. 


FIG.  58. 


If  there  are  three  points  of  contact,  the  base  is  a  triangle ;  if 
four,  a  quadrilateral,  &c. 

When  the  vertical  through  the  centre  of  gravity  (called  the 
line  of  direction)  falls  with- 
in the  base,  the  body  is 
supported;  if  without,  it  is 
not  supported.  In  the  body 
A  (Fig.  58)  the  force  of 
gravity  acts  in  the  line  G  F, 

and  there  are  lines  of  re- 

si  stance  on  both  sides  of 

G  F,  as  G  G  and  G  E,  so  that  the  body  cannot  turn  on  the  edge 
of  the  base,  without  rising  in  an  arc  whose  radius  is  G  C  or  G  E. 
But,  in  the  body  B,  there  is  resistance  only  on  one  side ;  and 
therefore,  if  the  force  of  gravity  be  resolved  on  G  C  and  a  perpen- 
dicular to  it,  the  body  is  not  prevented  from  moving  in  the  direc- 
tion of  the  latter,  that  is,  in  the  arc  whose  radius  is  G  C. 

If  the  line  of  direction  fall  at  the  edge  of  the-  base,  the  least 
force  will  overturn  it. 

83.  Different  Kinds  of  Equilibrium. — If  the  base  is  re- 
duced to  a  line  or  point,  then,  though  there  may  be  support,  there 
is  no  firmness  of  support ;  the  body  will  be  moved  by  the  least 
force.    But  it  is  affected  very  differently  in  different  cases. 

When  it  is  moved  from  its  position  of  support  and  left,  it  will 
in  some  cases  return  to  it,  pass  by,  and  return  again,  and  continue 
thus  to  vibrate  till  it  settles  in  its  place  of  support  by  friction  and 
other  resistances.  This  condition  is  called  stable  equilibrium. 

In  other  cases,  when  moved  from  its  position  of  support  and 
left,  it  will  depart  further  from  it,  and  never  recover  that  position 
again.  This  is  called  unstable  equilibrium. 

In  other  cases  still,  the  body,  when  moved  from  its  place  of 
support  and  left,  will  remain,  neither  returning 
to  it  nor  departing  further  from  it.  This  is  called  FlG- 

neutral  equilibrium. 

84.  Stable    Equilibrium. — Let    the   body 
(Fig.  59)  be  suspended  on  the  pivot  A.    This  is 
its  base  of  support.    While  the  centre  of  gravity 
is  below  A,  the  line  of  direction  EOF  passes 
through  the  base,  and  the  body  is  supported.    Let 
it  be  moved  aside,  and  the  centre  of  gravity  be 
left  at  G.    Let  G  R  represent  the  force  of  gravity, 
and  resolve  it  into  G  N.  on  the  line  A  G,  and  NR, 
or  G  B,  perpendicular  to  A  G.     G  N  is  resisted 
by  the  strength  of  A,  and  G  B  moves  the  centre 


THREE    KINDS    OF    EQUILIBRIUM. 


53 


FIG. 


of  gravity  in  the  arc  whose  radius  is  A  G.  Hence  the  body  swings 
with  accelerated  motion  till  the  centre  of  gravity  reaches  0,  where 
the  force  G  B  becomes  zero.  But  by  its  inertia,  the  body  passes 
beyond  that  position,  and  ascends  on  the  other  side,  till  the  retard- 
ing force  of  gravity  stops  it  at  g,  as  far  from  0  as  G  is.  It  then 
descends  again,  and  would  never  cease  to  oscillate  were  there  no 
obstructions. 

85.  Unstable  Equilibrium.— Next,  let  the  body  be  turned 
on  the  pivot  till  the  centre  -of  gravity  G  is  at  P,  above  A  (Fig.  60). 
Then,  as  well  as  when  G  is  below  A,  the  body  is 
supported,  because  the  line  of  direction  E  P  Ppasses 

through  the  base  A.  But  if  turned  and  left  ;m  the 
slightest  degree  out  of  that  position,  it  cannot  re- 
cover it  again,  but  will  depart  further  and  further 
from  it.  Let  G  R  represent  the  force  of  gravity, 
and  let  it  be  resolved  into  G  N,  acting  through  A, 
and  G  B  perpendicular  to  it.  The  former  is  re- 
sisted by  A ;  the  latter  moves  G  away  from  P,  the 
place  of  support.  If  the  body  is  free  to  revolve 
about  A,  without  falling  from  it,  the  centre  of 
gravity  will,  by  friction  and  other  resistances,  finally 
settle  below  A,  as  in  the  case  of  stable  equilibrium. 

86.  Neutral  Equilibrium.— Once  more,  suppose  the  pivot 
supporting  the  body  to  be  at  G,  the  centre  of  gravity ;  then,  in 
whatever  situation  the  body  is  left,  the  line  of  direction  passes 
through  the  base,  and  the  body  rests  indifferently  in  any  position. 

These  three  kinds  of  equilibrium  may  be  illustrated  also  by 
bodies  resting  by  curved  surfaces  on  a  horizontal  plane.  Thus,  if 
a  cylinder  is  uniformly  dense,  it  will  always  have  a  neutral  equi- 
librium, remaining  wherever  it  is  placed.  But  if,  on  account  of 
unequal  density,  its  centre  of  gravity  is  not  in  the  axis,  then  its 
equilibrium  is  stable,  when  the  centre  of  gravity 
is  below  the  axis,  and  unstable  when  above  it. 

In  general,  there  is  stable  equilibrium  when 
the  centre  of  gravity,  on  being  disturbed  in 
either  direction,  begins  to  rise  ;  unstable  when, 
if  disturbed  either  way,  it  begins  to  descend; 
and  neutral  when  the  disturbance  neither  raises 
nor  lowers  the  centre  of  gravity. 

87.  Questions  on  the  Centre  of  Grav- 
ity.- 

1.  A  frame  20  feet  high,  and  4  feet  in  diam- 
eter, is  racked  into  an  oblique  form  (Fig.  61),  B  F  c 


FIG.  61. 

E     D 


54  MECHANICS. 

till  it  is  on  the  point  of  falling ;  what  is  its  inclination  to  the 
horizon?  Ans.  78°  27'  47". 

2.  A  stone  tower,  of  the  same  dimensions  as  the  former,  is  in- 
clined till  it  is  about  to  fall,  but  preserves  its  rectangular  form ; 
what  is  its  inclination  ?  Ans.  78°  41'  24". 

3.  A  cube  of  uniform  density  lies  on  an  inclined  plane,-  and  is 
prevented  by  friction  from  sliding  down;  to  what  inclination 
must  the  plane  be  tipped,  that  the  cube  may  just  begin  to  roll 
down  ?  Ans.  45°. 

4.  What  must  be  the  inclination  of  a  plane,  in  order  that  a 
regular  prism  of  any  given  number  of  sides  may  be  at  the  limit 
between  sliding  and  rolling  down  ? 

Ans.  Equal  to  half  the  angle  at  the  centre  of  the  prism^ 
subtended  by  one  side. 

5.  A  body  weighing  83  Ibs.  is  suspended,  and  drawn  aside  from 
the  vertical  9° ;  what  pressure  is  there  on  the  point  of  support, 
and  what  force  urges  it  down  the  arc  ? 

Ans.  Pressure  on  the  support,  81.978  Ibs. 
Moving  force,  12.984  Ibs. 

88.  Motion  of  the  Centre  cf  Gravity  of  a  System 
when  one  of  the  Bodies  is  Moved. — 

When  one  body  of  a  system  is  moved,  the  centre  of  gravity  of  the 
system  moves  in  a  similar  path,  and  its  velocity  is  to  that  of  the  moving 
body  as  the  mass  of  that  body  is  to  the  mass  of  the  whole  system. 

If  the  system  contains  but  two  bodies,  A  and  B  (Fig.  62),  sup- 
pose A  to  remain  at  rest,  while  B 
describes  the  straight  lines  B  C, 
C  D,  &c.,  the  centre  of  gravity  G 
will  in  the  same  time  describe  the 
similar  series,  6r  H,  H  J,  &c. 
When  B  is  in  the  position  B,  and 
the  centre  of  gravity  at  G9  A  G  : 
A  B  : :  B  :  A  -f  .#;  when  B  is 
at  CyAHiACiiB-.A  +  B; 
.\AQ\AB\\AH\AC.  Hence 
G  H  is  parallel  to  B  C,  and  G  H:  B  C: :  B :  A  +  B.  In  like 
manner,  H  J:  CD  : :  B  :  A  +  B,  &c.  Thus,  all  the  parts  of  one 
path  are  parallel  to  the  corresponding  parts  of  the  other,  and  have 
a  constant  ratio  to  them.  Therefore  the  paths  are  similar.  As 
the  corresponding  parts  are  described  in  equal  times,  their  lengths 
are  as  the  velocities.  But  the  lengths  are  as  B :  A  +  B;  there- 
fore the  velocity  of  the  common  centre  of  gravity  is  to  that  of  the 
moving  body  as  the  mass  of  the  moving  body  is  to  the  mass  of  both. 
The  same  reasoning  is  applicable  when  the  body  moves  in  a  curve. 


MOTION     OF    CENTRE    OF    GRAVITY.  55 

If  the  system  contain  any  number  of  bodies,  and  the  centre  of 
gravity  of  the  whole  be  at  G,  then  the  centre  of  gravity  of  all 
except  B  must  be  in  the  line  B  G  beyond  G.  Suppose  it  to  be  at 
A,  and  to  remain  at  rest,  while  B  moves ;  then  it  is  proved  in  the 
same  manner  as  before,  that  Gr,  the  centre  of  gravity  of  the  whole 
system,  moves  in  a  path  parallel  to  the  path  of  B,  and  with  a 
velocity  which  is  to  JB's  velocity  as  the  mass  of  B  to  the  mass  of 
the  entire  system. 

89.  Motion  of  the  Centre  of  Gravity  of  a  System  when 
Several  of  the  Bodies  are  Moved. — 

When  any  or  all  of  the  bodies  of  a  system  are  moved,  the  centre 
of  gravity  moves  in  the  same  manner  as  if  all  the  system  were  collected 
there,  and  acted  on  by  the  forces  which  act  on  the  separate  bodies. 

Let  A,  B,  C,  &c.  (Fig.  63),  belong  to  a  system  containing  any 
number  of  bodies,  and  let  M  be  the  mass  of  the  system.  Let  A 
be  moved  over  A  a,  B  over  B  b,  G  over  G  c,  &c.  And  first  sup- 
pose the  motions  to  be  made  in  equal  successive  times.  If  the 
centre  of  gravity  of  the  system  is  first  at  G,  then  that  of  all  the 
bodies  except  A  is  in  A  G  produced,  as  at  g.  While  A  moves  to  a, 
G  moves  in  a  parallel  line  to  H  (Art.  88),  and  G  H :  A  a  : :  A  :  M. 
In  like  manner,  when  B  describes  B  b,  the  centre  of  gravity  of  the 
other  bodies  being  at  h,  the  centre  of  gravity  of  the  system  de- 
scribes the  parallel  line,  H  /i,  and  H  K :  B  b  : :  B  :  M ;  and  when 
C  moves,  K  L  :  C  c::  C :  M,  &c.  Now,  A  a  and  G  H  represent 
the  respective  velocities  of  the 
body  A,  and  the  system  M;  FIG.  63. 

therefore,  if  we  convert  the 
proportion  G  H:  A  a::  A:  M 
into  an  equation,  we  have  A  x 
A  a  =  M  x  G  H;  that  is,  the 
momentum  of  the  body  A 
equals  the  momentum  of  the 
system  M.  It  therefore  re- 
quires the  same  force  to  move 
A  over  A  a  as  to  move  the  system  M  over  G  H.  The  same  is  true 
of  the  other  bodies.  If  then  the  several  forces  which  move  the 
bodies,  limiting  the  number  to  three,  for  the  present,  were  applied 
successively  to  the  system  collected  at  G,  they  would  move  it  over 
G  H,  H  K,  K  L.  But  if  applied  at  once,  they  would  move  it  over 
G  L,  the  remaining  side  of  the  polygon.  If,  therefore,  the  forces, 
instead  of  acting  successively  on  the  bodies,  were  to  move  A  over 
A  a,  B  over  B  b,  and  G  over  C  c,  at  the  same  time,  the  centre  of 
gravity  of  the  system  would  describe  G  Lin  the  same  time.  In  the 
same  way  it  may  be  proved,  that  whatever  forces  are  applied  to  the 


56  MECHANICS. 

several  bodies  of  a  system,  the  centre  of  gravity  of  the  system  is 
moved  in  the  same  manner  as  a  body  equal  to  the  whole  system 
would  be  moved,  if  all  the  same  forces  were  applied  to  it. 

It  is  possible  that  the  centre  of  gravity  of  a  system  should 
remain  at  rest,  while  all  the  bodies  in  it  are  in  motion.  For,  sup- 
pose all  the  forces  acting  on  the  bodies  to  be  such  that  they  might 
be  represented  in  direction  and  intensity  by  all  the  sides  of  a  poly- 
gon, then,  since  a  single  body  acted  on  by  them  would  be  in  equi- 
librium, therefore  the  centre  of  gravity  of  the  system  would  remain 
at  rest,  though  the  bodies  composing  it  are  in  motion. 

90.  Mutual  Action  among  the  Bodies  of  a  System.— 

The  forces  which  have  b'een  supposed  to  act  on  the  several  bodies 
of  a  system  are  from  without,  and  not  forces  which  some  of  the 
bodies  within  the  system  exert  on  others.  If  the  bodies  of  a  sys- 
tem mutually  attract  or  repel  each  other,  such  action  cannot  affect 
the  centre  of  gravity  of  the  whole  system.  For  action  and  reac- 
tion are  always  opposite  and  equal.  Whatever  force  one  body 
exerts  on  any  other  to  move  it,  that  other  exerts  an  equal  force  on 
the  first,  and  the  two  actions  produce  equal  and  opposite  effects  on 
the  centre  of  gravity  between  them.  Therefore  the  centre  of 
gravity  of  a  system  remains  at  rest,  if  the  bodies  which  compose  it 
are  acted  on  only  by  their  mutual  attractions  or  repulsions. 

91.  Examples  on  the  Motion  of  the  Centre  of  Gravity. — 

1.  Two  bodies,  A  and  B,  of  given  weights,  start  together  from 
D  (Fig.  64),  and  move  uniformly  with  given  velocities  in  the  direc- 
tions D  A  and  D  B ;  required  the  di- 
rection and  velocity  of  their  centre  of  FIG.  64. 
gravity. 

As  the  directions  of  D  A  and  D  B 
are  given,  we  know  the  angle  A  D  B] 
from  the  given  velocities,  we  also  know 
the  lines  D  A  and  D  B,  described  in  a 
certain  time.  Calculate  the  side  A  B, 
and  the  angles  A  and  B.  Find  the 

place  of  the  centre  of  gravity  G  between  the  bodies  at  A  and  B. 
Then,  in  the  triangle  D  B  G,  D  B,  B  G,  and  angle  B  are  known, 
by  which  may  be  found  the  distance  D  G  passed  over  by  the  cen- 
tre of  gravity  in  the  time,  and  B  D  G  the  angle  which  its  path 
makes  with  that  of  the  body  B. 

2.  The  bodies  A  and  B,  of  given  weights,  start  together  from  D 
(Fig.  65),  and  move  with  equal  velocities  in  opposite  directions 
around  the  circumference  of  a  circle,  meeting  again  at  D ;  what  is 
the  path  of  their  centre  of  gravity  ? 

Draw  the  diameter,  D  E,  and  join  A  and  B,  the  points  which 


MOTION  OF  CENTRE  OF  GRAVITY. 


the  bodies  have  reached  after  any  given  time.  As  D  A  and  D  B 
are  equal  arcs,  A  B  is  perpendicular  to  D  E,  and  is  bisected  by  it 
Let  G  be  the  common  centre  of  gravity  of  A  and  B,  then 


AiBn  B  G:A 
A+B:A-B::BG  + 

::        AB 

::        AN 


BG-A 
G  J/; 
GN. 


FIG.  65. 


Therefore  A  N9  the  ordinate  of  the  circle,  is  to  G  N,  the  corre- 
sponding ordinate  of  the  figure  de- 
scribed by  the  centre,  always  in  the 
same  constant  ratio,  of  the  sum  of 
the  bodies  to  their  difference.  But 
this  is  a  property  of  the  ellipse,  that, 
when  its  axis  is  the  diameter  of  a 
circle,  the  corresponding  ordinates 
of  the  two  figures  are  in  a  constant 
ratio.  Hence  the  centre  of  gravity 
of  A  and  B  describes  an  ellipse, 
while  they  move,  in  the  manner 
before  stated,  round  the  circle. 

If  the  bodies  approach  equality, 

their  difference  grows  less,  and  therefore  the  ellipse  more  eccentric, 
till,  when  the  bodies  are  equal,  the  path  of  the  centre  of  gravity  is 
a  straight  line,  as  it  evidently  should  be,  in  order  to  bisect  the 
chords,  A  B,  b  a,  &c. 

3.  Three  bodies  of  given  weight,  A,B,  G,  in  the  same  time  and 
in  the  same  order,  describe  with  uniform  velocity  the  three  sides 
of  the  given  triangle  A  B-G  (Fig.  66) ;  required  the  path  of  their 
centre  of  gravity. 

Let  G  be  their  centre  of 
gravity  before  they  move.  If 
they  move  successively,  G  de- 
scribes G  K,  K  L,  L  M,  par- 
allel to  the  sides  of  the  trian- 


FIG.  66. 


gle,  and  having  to  them  re- 
spectively the  same  ratios  as 
the  corresponding  moving 
bodies  have  to  the  sum  of  the 
bodies  (Art.  89).  Thus,  three 

sides  of  the  polygon  are  known ;  and  the  angle  K=  B,  and  L  =  C. 
These  data  are  sufficient  for  calculating  the  fourth  side,  G  M,  which 
the  centre  of  gravity  describes,  when  the  bodies  move  together. 

4.  Show  that  when  the  three  bodies  in  Example  3d  are  equal, 
the  centre  of  gravity  will  remain  at  rest. 


58 


MECHANICS. 


5.  A  (Fig.  67)  weighs  one  pound; 
B  weighs  two  pounds,  and  lies  direct- 
ly east  of  A\  they  move  simulta- 
neously, A  northward,  and  B  east- 
ward, at  the  same  uniform  rate  of  40 
feet  per  second ;  required  the  direc- 
tion and  yelocity  of  their  centre  of 
gravity. 


FIG.  67. 


Ans.  Velocity  is  29.814  feet  per  second. 
Direction  is  E.  26°  33'  54"  N. 


CHAPTER  V. 

THE    COLLISION    OF    BODIES. 

92.  Elastic   and    Inelastic    Bodies.— Mastic  bodies  are 
those  which,  when  compressed,  or  in  any  way  altered  in  form, 
tend  to  return  to  their  original  state.     Those  which  show  no  such 
tendency  are  called  inelastic  or  non-elastic.     No    substance  is 
known  which  is  entirely  destitute  of  the  property  of  elasticity ; 
but  some  have  it  in  so  small  a  degree,  that  they  are  called  inelas- 
tic, such  as  lead  and  clay.    Elasticity  is  perfect  when  the  restoring 
force,  whether  great  or  small,  is  equal  to  the  compressing  force. 
Air,  and  the  gases  generally,  seem  to  be  almost  perfectly  elastic ; 
ivory,  glass,  and  tempered  steel,  are  imperfectly,  though  highly, 
elastic ;  and  in  different  substances,  the  property  exists  in  all  con- 
ceivable degrees  between  the  above-named  limits. 

93.  Mode  of  Experimenting. — Experiments  on  collision 
may  be  made  with  balls  of  the 

same    density    suspended    by  FIG.  68 

long  threads,  so  as  to  move  in 
the  line  which  joins  their  cen- 
tres of  gravity.  If  the  arcs 
through  which  they  swing  are 
short  compared  with  their 
radii,  the  balls,  let  fall  from 
different  heights,  will  reach  the 
bottom  sensibly  at  the  same 
time,  and  will  impinge  with 
velocities  which  are  very  nearly 
proportional  to  the  arcs.  Thus 
A  (Fig.  68),  falling  from  6, 


COLLISION    OF    INELASTIC    BODIES.  59 

and  B  from  3,  will  come  into  collision  at  0,  with  velocities  which 
are  as  2  :  1. 

94.  Collision  of  Inelastic  Bodies.—  Such  bodies,  after  im- 
pact, move  together  as  one  mass. 

The  velocity  of  two  inelastic  bodies  after  collision  is  equal  to  the 
algebraic  sum  of  their  momenta,  divided  by  the  sum  of  the  bodies. 

Let  A,  B,  represent  the  masses  of  the  two  bodies,  and  a,  b,  their 
respective  velocities.  Considering  a  as  positive,  if  B  moves  in  the 
opposite  direction,  its  velocity  must  be  called  —  b.  Let  v  be  the 
common  velocity  after  impact. 

1.  Same  directions.  —  The  momentum  of  A  is  A  a;  that  of  B  is 
B  b]  and  the  momentum  of  both  after  collision  is  (A  +  B)  v. 
According  to  the  third  law  of  motion  (Art.  13),  whatever  mo- 
mentum A  loses,  B  gains,  so  that  the  whole  momentum  is  the 
same  after  collision  as  before  ;  therefore 

Aa  +  Bl>  =  (A  +  B)v;.:v  =  -  ?*f  * 

.0.  ~j~  Jj* 

A  a 

If  B  is  at  rest  before  impact,  5=0,  and  v  —  -7-r~R' 

A  ~\~  JD 

To  find  the  loss  or  gain  of  velocity  for  either  body,  multiply 
the  other  body  by  the  difference  of  velocities,  and  divide  fey  the  sum 
of  the  bodies.  For,  A's  velocity  before  impact  was  a  ;  after  impact 
...  Aa  +  Bb  ,.  Aa  +  Bb  B(a-b) 

Jt  1S  -    theref°re  the  l°SS  =  a  ~    ---  - 


But  B's  gain  is  the  velocity  after  impact  diminished  by  the  velo- 

Aa  +  Bb  A(a  —  b) 

city  before,  ^———  -  b  =  -. 


When  B  is  at  rest,  these  expressions  become 

B  a     „        .,  Aa     „     r.. 

A   ,    r>  i°r  •"  loss  ;  —A  --  5  for  B's  gam. 

A.   +   £>  A.   +  £> 

2.  Opposite  directions.  —  Since  b  is  negative,  v  =  —A  -  ^—  . 

A  -f-  Jj» 

To  find  loss  or'  gain  in  this  case,  multiply  the  other  body  by  the 
sum  of  the  velocities,  and  divide  by  the  sum  of  the  bodies.  For, 
A9  s  loss 

_  Aa-JW  _  B(a  +  b) 
A+B       '   A  +  B  9 
and  B's  gain 

-  A  a  ~  B  b  _  (—  M  —  Aa  —  Bb      ,  _  A(a  +  b) 
A+B  A+B  :   A+B 

In  the  case  of  opposite  motions,  the  formula  for  v  becomes  zero, 
when  A  a  =  B  b  ;  but  in  that  case,  A  :  B  :  :  b  :  a.  Hence,  if 
bodies  which  meet  each  other  have  velocities  inversely  as  the 
quantities,  they  will  be  at  rest  after  the  collision. 


60  MECHANICS. 

95.  Questions  on  Inelastic  Bodies.— 

.1.  A,  weighing  3  oz.,  and  moving  10  feet  per  second,  overtakes 
B,  weighing  2  oz.,  and  moving  3  feet  per  second ;  what  is  the 
common  velocity  after  impact?  Am.  7^  feet  per  second. 

2.  A  weight  of  7  oz.?  moving  11  feet  per  second,  strikes  upon 
another  at  rest  weighing  15  oz. ;  required  the  velocity  after  im- 
pact ?  Ans.  3-J-  feet  per  second. 

3.  A  weighs  4  and  B  %  pounds ;  they  meet  in  opposite  direc- 
tions, A  with  a  velocity  of  9,  and  B  with  one  of  5  feet  per  second ; 
what  is  the  common  velocity  after  impact  ? 

Ans.  4-J  feet  per  second. 

4.  A  =  7  pounds,  B  =  4  pounds ;  they  move  in  the  same  di- 
rection, with  velocities  of  9  and  2  feet  per  second ;  required  the 
velocity  lost  by  A  and  gained  by  B?  Ans.  A  2T6T,  B  4T5T. 

5.  A  body  moving  7  feet  per  second,  meets  another  moving 
3   feet  per  second,  and  thus    loses  half  its  momentum;    what 
are  the  relative  masses  of  the  two  bodies  ? 

Ans.  A:  B::  13  it. 

6.  A  weighs  6  pounds  and  B  5 ;  B  is  moving  7  feet  per  sec- 
ond, in  the  satae  direction  as  A ;  by  collision  JB's  velocity  is 
doubled ;  what  was  A's  velocity  before  impact  ? 

Ans.  19 1  feet  per  second. 

96.  Collision  of  Elastic  Bodies. — Elastic  bodies  after  col- 
lision do  not  move  together,  but  each  has  its  own  velocity.    These 
velocities  are  found  by  doubling  the  loss  and  gain  of  inelastic 
bodies.     When  the  elastic  body  A  impinges  on  B,  it  loses  velocity 
while  it  is  becoming  compressed^  and  again,  while  recovering  its 
form,  it  loses  as  much  more,  because  the  restoring  force  is  equal  to 
the  compressing  force.    For  a  like  reason,  B  gains  as  much  velo- 
city while  recovering'  its  form  as  it  gained  while  being  compressed 
by  the  action  of  A.    Hence,  doubling  the  expressions  for  loss  and 
gain  given  in  Art.  94,  and  applying  them  to  the  .original  velocities, 
we  find  the  velocity  of  each  body  after  collision,  on  the  supposition 
of  perfect  elasticity. 

When  the  directions  are  the  same, 

2  B  (a  -  I) 
the  velocity  of  A  =  a ~T \  n     > 

,,        „  0      ,      2  A  (a  —  I) 
that  of  B  =  I  H j      p    . 

j~L  +  £>• 

When  the  directions  are  opposite, 

,,                                      %  B  (a  4-  b) 
the  velocity  of  A  =  a ~ — ^~- ; 

2  A  (a  +  5) 

that  of  B  —  —  I  +  — r^— rr-^ 
A  +  B 


COLLISION    OF    ELASTIC    BODIES.  gl 

Reducing  these  expressions,  we  have  for  the  velocities  of  elastic 
bodies  after  collision  the  following  formulee  : 

(1.)  Same  direction.    Velocity  of  A  =  i^L_^±+M^. 


TT  ,    .. 
(2.)  Same  airection.    Velocity  of  B  = 


» 


(3.)  Opposite  directions.    Velocity  of  A  =  -  ~  —  ^  —  —  . 

J±    4"    -O 

(4.)  Opposite  directions.    Velocity  of  .Z?  =  -  -  ~  —  ~  —  —  . 

^l  H-  x) 

97.  Equal  Elastic  Bodies.  —  After  the  impact  of  equal  elastic 
bodies,  each  takes  the  original  velocity  of  the  other.    When  A—B, 
formula  (1)  is  reduced  to  b;  and  formula  (2)  to  a\  that  is,  A  has 
IPs  former  velocity,  and  B  has  A's.   The  same  is  true  if  they  move 
in  opposite  directions.    For,  when  A  =  B,  formula  (3)  becomes 
—  b,  which  was  B's  original  velocity,  and  formula  (4)  becomes  a, 
which  was  A's.    Therefore,  in  the  opposite  motions  of  equal  elastic 
bodies,  collision  causes  each  to  rebound,  since  +  a  is  exchanged 
for  —  b,  and  —  b  for  4-  a. 

If  we  reduce  these  four  formulae  for  the  case  in  which  A  —  B, 
and  B  is  at  rest,  we  find  the  same  interchange  of  conditions  ;  for 
formula  (1)  becomes  0,  and  formula  (2)  becomes  a;  so  formula  (3) 
becomes  0,  and  formula  (4)  becomes  a. 

98.  Unequal  Elastic  Bodies.  — 

1.  If  a  greater  body  impinge  on  a  less  one  at  rest,  the  imping- 
ing body  goes  forward,  but  slower  than  before,  and  the  other  pre- 
cedes it  with  a  greater  velocity  than  the  impinging  body  first  had. 

For,  formula  (1)  becomes  ^—3  --  ~  ,  which  is  positive,  but  less 

j?l  -p  -tj 

than  a  \  therefore  it  advances,  though  slower  than  before.    But 

2  A  a 
formula  (2)  becomes  -.  -  ^,  which  is  greater  than  a  ;  hence,  I> 

goes  on  faster  than  A  did  before  collision. 

2.  If  a  less  body  impinge  on  a  greater  one  at  rest,  it  rebounds, 
and  the  other  goes  forward,  but  with  a  less  velocity  than  that 

which  the  impinging  body  first  had.    For,  ±—-i  --  ~  is  negative, 

u.'l  ~r  Jj 

,  %Aa  .   ,       ,, 
and  -T  —  -£  is  less  than  a. 

3.  If  two   elastic  bodies,  having  equal  velocities,  meet  each 
other,  and  one  of  them  is  brought  to  rest,  its  mass  is  three  times 
as  great  as  that  of  the  other.    For.  as  the  velocities  are  equal,  by 


MECHANICS. 


formula  (3),  (A  ~-  =  0;  /.  (A  -  B)  a  -  2  Ba  =  0; 


FIG.  69. 


99.  Series  of  Elastic  Bodies.  — 

1.  Equal  bodies.  —  Let  a  row  of  equal  elastic  bodies,  A,  B,  C.  .  . 
(Fig.  69)  be    suspended  in  contact;    then 

(Art.  97),  if  A  be  drawn  back  and  left  to 
fall  against  B,  it  will  rest  after  impact,  and 
B  will  tend  to  move  on  with  As  velocity  ; 
after  the  impact  of  B  on  (7,  B  will  remain, 
and  Cf  tend  to  move  with  the  same  velocity  ; 
and  so  the  motion  will  be  transmitted  through 
the  series,  and  F  will  move  away,  while  all 
the  others  remain  at  rest. 

2.  Decreasing  series.  —  If  the  bodies  de- 
crease, as  A,  B,  C,  &c.  (Fig.  70),  and  A  be 
drawn  back  to  A,  and  allowed  to  fall  against 
£y  then  (Art.  98)  A  still  moves  forward, 
while  B  receives  a  greater  velocity  than  A 

had,  C  still  greater,  &c.  The  last  of  the  series,  therefore,  moves 
with  the  greatest  velocity,  and  each  one  with  a  greater  velocity 
than  that  which  impinged  on  it. 


3.  Increasing  series. — If  the  bodies  increase,  as  A,  B,  C,  &c. 
(Fig.  71),  then,  when  A  falls  from  A  against  B,  it  imparts  to  B  a 


FIG.  71. 


less  velocity  than  it  had  itself,  and  rebounds  (Art  98) ;  in  like 
manner  B  rebounds  from  C,  and  so  on ;  while  the  last  of  the  series 
goes  forward  with  less  velocity  than  any  previous  one  would  have 
had  if  it  had  been  the  last. 


EFFECT    OF    COLLISION    ON    LIVING    FORCE.     63 
If  the  bodies  in  Fig.  70  are  in  geometrical  progression,  the 

(O        \  n  —  1 
i / 

Let  the  series  be  A,  Ar,  Ar* ....  Arn  ~  \ 

By  Art.  98,  when  A  impinges  on  B  at  rest  the  velocity  com- 

2  Aa          2  Aa  2a 

mumcated  to  B  is  ^ =  =  -, -r—  =  :T3T~  ~  &• 

Again,  the  velocity  imparted  to  G  is 
2B1>  2  Ar 

B  +  G~  Ar  +  Ar* 

Hence  the  successive  velocities  are  a,  T    --,  ,  ~      X2,  &c.,  from 


which  it  appears  that  any  term  in  the  series  is  found  by  multiply- 
ing the  original  velocity  by  2,  raised  to  a  power  one  less  than  the 
number  of  terms  '  and  divided  by  1  +  r  raised  to  the  same  power. 

O«  —    1     £, 

Consequently,  the  last  term  is  -T-  -  —  ;T.    Hence,  vel.  of  the  first  : 


2n~la  i    2 


i 
:  I  — 

\1 


vel.  of  the  last  :  :  a  : 


100.  Questions  on  Elastic  Bodies.  — 

1.  A,  weighing  10  Ibs.  and  moving  8  feet  per  second,  impinges 
on  B,  weighing  6  Ibs.  and  moving  in  the  same  direction,  5  feet 
per  second  ;  what  are  the  velocities  of  A  and  B  after  impact  ? 

Ans.  A's  =  5f  ,  B's  =  8|. 

2.  A  :  B  :  :  4  :  3  ;  directions  the  same  ;  velocities  5:4;  what  is 
the  ratio  of  their  velocities  after  impact  ?  Ans.  29  :  36. 

3.  A,  weighing  4  Ibs.,  ^locity  6,  meets  B,  weighing  8  Ibs., 
velocity  4  ;  required  their  respective  directions  and  velocities  after 
collision  ?  Ans.  A  is  reflected  back  with  a  velocity  of  7  J, 

and  B  with  a  velocity  of  2f  . 

4.  A  and  B  move  in  opposite  directions  ;  A  equals  4  B,  and 
b  =  2  a  ;  how  do  the  bodies  move  after  collision  ? 

Ans.  A  returns  with  J,  B  with  If  its  original  velocity. 

5.  There  are  ten  bodies  whose  masses  increase  geometrically 
by  the  constant  ratio  3,  and  the  first  impinges  on  the  second 
with  the  velocity  of  5  feet  per  second  ;  required  the  motion  of  the 
last  body  ?  Ans.  The  last  body  would  move  with  the 

velocity  of  -f  2  feet  per  second. 

101.  Living  Force  lost  in  the  Collision  of  Inelastic 

Bodies.—  The  amount  of  living  force  (Art.  35)  before  collision  is 

A  a?  +  B  V  ;  and  after  collision  it  is  (A  +  B)  x 


jo.  ~r  Jo 


Subtract  the  latter  from  the  former,  and  call  the 


64  MECHANICS. 

remainder  d.    Then  d  =  A  a?  -f  B  V—  -  —  -,  --  7?~'    Expanding 

and  uniting  terms,  d  =  —  ~  —  /?"•    This  value  of  d  is  positive, 

because  (a  —  #)2  is  necessarily  positive,  as  well  as  A  and  B.  There- 
fore there  is  always  a  loss  of  living  force  in  the  collision  of  inelas- 
tic bodies. 

102.  Living  Force  Preserved  in  the  Collision  of  Elastic 
Bodies.  —  The  living  force  of  A  before  collision  is  A  a*  ;  after  col- 

lision, it  is  A  x  —  --  -  '  a     ^  —  -.  Subtracting  the  latter  from 
(A  +  ±>) 

the  former,  the  loss  (supposing  there  is  loss)  is 

(A  +  J3YAa'2-(A-£YA^-4:(A--  B}  AB  ab-^AB*  tf 

(A  +  BY 
The  living  force  of  B  before  collision  is  B  ft*  ;  after  collision,  it 

.    _      {(B-A)b  +  2  A  a}* 

is  B  x  -  --  -r^-  —  -g-  ---  ;  and  the  expression  for  loss  is 


Therefore  the  total  loss  of  living  force  is  the  sum  of  the  expres- 
sions (1.)  and  (2.). 

Reducing  the  two  first  terms  in  each  fraction  to  one,  the  frac- 
tions become 

4  A"  B  a?  -  4  (A  -  B)  A  Ba  I  -  4  A  E>  tf 

(A  +  BY  *  '  '  '  W 


and  (A  +}T-  -  ....  (4.) 

If  the  fractions  (3)  and  (4)  be  addid,  it  is  evident  that  the  nu- 
merators cancel  each  other,  and  therefore  the  sum  of  the  fractions 
is  zero.  Hence,  there  is  no  loss  of  living  force  in  the  collision  of 
elastic  bodies. 

103.  Impact  on  an  Immovable  Plane.  —  If  an  inelastic 
body  strikes  a  plane  perpendicularly,  its  motion  is  simply  destroyed; 
in  strictness,  however,  it  imparts  an  infinitely  small  velocity  to  the 
body  called  immovable.  If  it  strikes  obliquely,  and  the  plane  is 
smooth,  it  slides  along  the  plane  with  a  diminished  velocity.  Let 
A  L  (Fig.  72)  represent  the  motion  of  the  body  before  impact  on 
the  plane  P  N,  and  resolve  it  into 
A  C,  perpendicular,  and  C  L,  par-  FlG- 

allel  to  the  plane.  Then  A  C,  as 
before,  is  destroyed,  but  C  L  is 
not  affected;  hence  the  former 
velocity  is  to  its  velocity  on  the 
plane,  as  A  L  :  C  L  ::  radius  :  co- 
sine of  the  inclination. 


CLASSIFICATION   OF   MACHINES.  C5 

If  a  perfectly  elastic  body  impinges  perpendicularly  upon  a 
plane,  then,  after  its  motion  is  destroyed,  the  force  by  which  it 
resumes  its  form  causes  an  equal  motion  in  the  opposite  direction ; 
that  is,  the  body  rebounds  in  its  own  path  as  swiftly  as  it  struck. 
But  if  tlie  impact  is  oblique,  the  body  rebounds  at  an  equal  angle 
on  the  opposite  side  of  the  perpendicular.  For,  resolve  A  L,  as 
before,  into  A  C,  C  L ;  the  latter  continues  uniformly;  but,  instead 
of  the  component  A  C,  there  is  an  equal  motion  in  the  opposite 
direction.  Therefore,  if  L  D  is  made  equal  to  C  L,  and  D  E  equal 
to  A  C,  the  resultant  ofLD  and  D  E  is  L  E,  which  is  equal  to 
A  L,  and  has  the  same  inclination  to  the  plane.  Hence,  the 
angles  of  incidence  and  reflection  are  equal,  and  on  opposite  sides 
of  the  perpendicular  to  the  surface  at  the  place  of  impact. 

104.  Imperfect  Elasticity. — The  formulae  for  the  velocity 
of  bodies  after  collision,  and  the  statements  of  the  preceding  arti- 
cle, are  correct  only  on  the  supposition  that  bodies  are,  on  the  one 
hand,  entirely  destitute  of  elasticity,  or  on  the  other  perfectly 
elastic.  As  no  solid  bodies  are  known,  which  are  strictly  of  either 
class,  these  deductions  are  found  to  be  only  near  approximations 
to  the  results  of  experiment.  In  all  practical  cases  of  the  impact 
of  movable  bodies,  the  loss  and  gain  of  velocity  are  greater  than  if 
they  were  inelastic,  and  less  than  if  perfectly  elastic.  And  in  cases 
of  impact  on  a  plane,  there  is  always  some  velocity  of  rebound,  but 
less  than  the  previous  velocity ;  and  therefore,  if  the  collision  is 
oblique,  the  body  has  less  velocity,  and  makes  a  smaller  angle  with 
the  plane  than  before.  For,  making  D  F  less  than  A  C,  the 
resultant  L  F  is  less  than  A  L,  and  the  angle  D  L  F  is  smaller 
than  D  L  E,  or  A  L  6'. 


CHAPTER   VI. 

SIMPLE     MACHINES. 

105.  Classification  of  Machines. — In  the  preceding  chap- 
ters, the  motion  of  bodies  has  been  supposed  to  arise  from  the  im- 
mediate action  of  one  or  more  forces.  But  a  force  may  produce 
effects  indirectly,  by  means  of  something  which  is  interposed  for 
the  purpose  of  changing  the  mode  of  action.  These  intervening 
bodies  are  called,  in  general,  machines ;  though  the  names,  tools, 
instruments,  engines,  &c.,  are  used  to  designate  particular  classes 
of  them.  The  elements  of  machinery  are  called  simple  machines. 
The  following  list  embraces  those  in  most  common  use: 

1.  The  lever. 

2.  The  wheel  and  axle. 

5 


66  MECHANICS. 

3.  The  pulley. 

4.  The  rope  machine. 

5.  The  inclined  plane. 

6.  The  wedge.      ^ 

7.  The  screw. 

8.  The  knee-joint. 

In  respect  to  principle,  these  eight,  and  all  others,  may  be  re- 
duced to  three. 

1.  The  law  of  equal  moments,  applicable  in  those  cases  in  which 
the  machine  turns  on  a  pivot  or  axis,  as  in  the  lever  and  the  wheel 
and  axle. 

2.  The  principle  of  transmitted  tension,  to  be  applied  wherever 
the  force  is  exerted  through  a  flexible  cord,  as  in  the  pulley  or 
rope  machine. 

3.  The  principle  of  oblique  action,  applicable  to  all  the  other 
machines,  the  force  being  employed  to  balance  or  overcome  one 
component  only  of  the  resistance. 

The  force  which  ordinarily  puts  a  machine  in  motion  is  called 
the  power;  the  force  which  resists  the  power,  and  is  balanced  or 
overcome  by  it,  is  called  the  weight. 

A  compound  machine  is  one  in  which  two  or  more  simple  ma- 
chines are  so  connected  that  the  weight  of  the  first  constitutes  the 
power  of  the  second,  the  weight  of  the  second  the  power  of  the 
third,  &c. 

I.  THE  LEVEK. 

106.  The  Three  Orders  of  Straight  Lever. — The  lever  is 
a  bar  of  any  form,  free  to  turn  on  a 
fixed  point,  which  is  called  the  fulcrum.  FlG- 73-  c         B 

In  the  first  order  of  lever,  the  fulcrum    '    ,- ^ . 

is  between  the  power  and  weight  (Fig.    PB  F 

73) ;  in  the  second,  the  weight  is  between  Q> 

the  power  and  fulcrum  (Fig.  74) ;  in 

the  third,  the  power  is  between  the  weight  and  fulcrum  (Fig.  75). 

FIG.  74  FIG.  75. 


If  P  and  W,  in  either  of  these  figures,  represent  forces  acting 


EQUAL    MOMENTS.  67 

in  vertical  lines,  then  the  circumstances  of  equilibrium  are  deter- 
mined by  the  laws  of  parallel  forces  (Art.  54).  In  Fig.  73,  if  P 
and  W  are  in  equilibrium,  their  resultant  will  be  so  situated  at  C9 
that  P  :  W  : :  B  C :  A  C',  and  the  fulcrum  must  be  at  that  point, 
and  be  able  to  sustain  a  pressure  equal  to  P  +  W.  In  Fig.  74,  P 
and  the  reaction  of  Fat  C  are  two  upward  forces,  whose  resultant 
is  counterbalanced  by  W",  then  W  is  represented  by  the  whole 
line,  and  P  by  the  part  B  G\  .-.  P:  W : :  B  C:  A  C,  as  before. 
The  pressure  on  F  equals  W  —  P.  In  Fig.  75,  W  and  the  reac- 
tion of  F  are  downward  forces,  whose  resultant  is  at  A,  in  equi- 
librium with  P.  Here  P  is  represented  by  the  whole  line  B  C, 
and  W  by  the  part  A  (7;  /.  P  :  W  : :  B  C  :  A  0.  The  upward 
pressure  against  F  is  equal  to  P  —  W. 

Hence,  in  each  order  of  the  straight  lever,  when  the  forces  act 
in  parallel  lines, 

The  power  and  weight  are  inversely  as  the  lengths  of  the  arms 
on  which  they  act. 

107.  Equal  Moments  in  Relation  to  tlie  Fulcrum. — 
Changing  the  proportion  into  an  equation,  we  find  for  each  order 
of  the  lever,  P  x  A  O=W  x  B  G\  that  is, 

The  power  and  weight  have  equal  moments  in  relation  to  the 
fulcrum. 

The  moment  of  either  force  is  the  measure  of  its  efficiency  to 
turn  the  lever ;  for,  since  the  lever  is  in  equilibrium,  the  efficiency 
of  the  power  to  turn  it  in  one  direction  must  equal  the  efficiency 
of  the  weight  to  turn  it  in  the  opposite  direction.  "We  may  there- 
fore use  P  x  A  C  to  represent  the  former,  and  W  x  B  C,  the 
latter. 
.  If  several  forces,  as  in  Fig.  76,  are  in  equilibrium,  some  tending 

FIG.  7G. 

A                B                      c  r> 
®— -- 9 


to  turn  the  bar  in  one  direction,  and  others  in  the  opposite,  then 
A  and  B  must  have  the  same  efficiency  to  produce  one  motion  as 
C  and  D  have  to  produce  the  opposite ;  that  is,  Ax  A  G+BxB  G 
=  Ox  C  G  +  D  x  DG\  or, 

The  sum  of  the  moments  of  A  and  B  equals  the  sum  of  the  mo- 
ments of  C  and  D. 

In  order  to  allow  for  the  influence  of  the  weight  of  the  lever 
itself,  consider  it  to  be  collected  at  its  centre  of  gravity,  and  add 
its  moment  to  that  of  the  power  or  weight,  according  as  it  aids 
the  one  or  the  other.  In  Fig.  73,  let  the  weight  of  the  lever  =  w, 


68  MECHANICS. 

and  the  distance  of  its  centre  from  G  on  the  side  of  P  =  m ;  then 
P  x  A  C  +  m  10  =  W  x  B  C.  In  the  2d  and  3d  orders,  the  mo- 
ment of  the  lever  necessarily  aids  the  weight ;  and  hence,  in  each 
case,  P  x  A  G=  W  x  B  C  +  mw. 

If  a  weight  hangs  on  a  bar  between  two  supports,  as  in  Fig.  77, 
it  may  be  regarded  as  a  lever  of  the 
2d  order,  the  reaction  of  either  sup-  FlG-  ?7- 

port  being  considered  as  a  power. 
Let  F  denote  the  reaction  at  A,  and 
F'  at  C;  then  by  the  theorems  of 
parallel  forces,  we  have  the  pressures 
at  A  and  C  inversely  as  their  dis- 
tances from  B,  and  W  —  F  +  F'. 

108.  The  Acting  Distance.— In  the  three  orders,  as  above 
described,  the  equilibrium  is  not  destroyed  by  inclining  the  lever 
to  any  angle  whatever  with  the  horizon,  provided  the  centre  of 
motion  G  is  at  the  centre  of  gravity  of  the  bar,  and  not  above  or 
below  it,  and  provided  the  directions  of  the  forces  remain  vertical. 
For,  by  the  principle  of  parallel  forces,  any  straight  line  intersect- 
ing the  lines  of  the  forces  is  divided  by  the  line  of  the  resultant 
into  -parts  which  are  inversely  as  the  forces;  therefore  (Fig.  78) 
I  G :  a  C : :  P  :  W.  Hence,  the  re- 
sultant of  P  and  W  remains  at  C,  in  FlG-  78- 


every  position  of  the  lever.    By  sim-      A  ML c^^^"^    B 

ilar  triangles,  I  G\  aG  \\G  N  \  G  M\ 

.-.  P-.W::  C  N:  CM.-,  .-.  P  x  CM 

=  W  x  CN.     The  lines   CM  and 

C  N,  which  are  drawn  from  the  ful-  w" 

crum  perpendicular  to  the  lines  in 

which  the  forces  act,  are  called  the 

acting  distances  of   the  power  and 

weight,  respectively.    And  as  they  may  be  employed  in  levers  of 

irregular  form,  the  moments  of  power  and  weight  are  usually 

measured  by  the  products,  P  x  C  Jfand  W  x  G  N;  therefore,  the 

power  multiplied  by  its  acting  distance  equals  the  weight  multiplied 

by  its  acting  distance  ;  or,  more  briefly,  the  moment  of  the  power 

equals  the  moment  of  the  weight,  as  in  Art.  107.    In  Figs.  73,  74, 

and  75,  the  acting  distances  are  in  each  case  identical  with  the 

arms  of  the  lever. 

109.  Lever  not  Straight,  and  Forces  not  Parallel.— 
Let  ACS  (Fig.  79)  be  a  lever  of  any  form,  and  let  it  be  in  equi-, 
librium  by  the  forces  P  and  P',  acting  in  any  oblique  directions 
in  the  same  plane.    Produce  P  A  and  P'  B  till  they  meet  in  Z>; 
then,  if  the  fulcrum  is  at  C,  the  resultant  must  be  in  the  direction 


THE    COMPOUND    LEVER. 


69 


FIG.  79. 


D  C;  otherwise  the  reaction  of  the  fulcrum  cannot  keep  the  sys- 
tem   in    equilibrium    (Art.  60,  2). 
Therefore  (Art.  61)  P :  P' :  :  sin 
B  D  C :  sin  A  D  C. 

Draw    G  M   perpendicular  to 
A  D,  and  C  N  to  B  D,  and  they 
are  the  sines  of  A  D  C  and  B  D  C, 
to  the  same  radius  D  C. 
.'.P:  P'::  C  N'.  CM;  andPx  CM 

=  P'  x  GN. 

The  lines  C  M  and  C  N  are  the  act- 
ing distances  of  P  and  P';  there- 
fore the  law  of  the  lever  in  all  cases  is  the  same,  namely : 

The  moment  of  the  power  equals  the  moment  of  the  weight. 

When  the  forces  act  obliquely,  the  pressure  on  the  fulcrum  is 
less  than  the  sum  of  the  forces ;  for,  if  C  E  is  parallel  to  B  D, 
then  D  E,  E  C,  and  G  D,  represent  the  three  forces  which  are  in 
equilibrium.  But  C  D  is  less  than  the  sum  of  D  E  and  E  C. 

110.  The  Compound  Lever. — When  a  lever  acts  on  a 
second,  that  on  a  third,  &c.,  the  machine  is  called  a  compound 
lever.  The  law  of  equilibrium  is — 

The  power  is  to  the  weight  as  the  product  of  the  acting  distances 
on  the  side  of  the  iveight  is  to  the  product  of  the  acting  distances  on 
the  side  of  the  power. 

Let  the  force  exerted  by  A  B  on  B  D  (Fig.  80)  be  called  x,  and 

FIG.  80. 


that  of  B  D  on  D  E  be  called  y ;  then 

P  x  BC-.AC; 
and  x  y  DF\BF\ 
and  y  W  E  G  :  D  G. 

Compounding  these  proportions,  and  dividing  the  first  couplet  by 
the  common  factors,  we  have 

P:W::BCxDFxEG:ACx£FxDG. 
If  the  levers  were  of  irregular  forms,  the  acting  distances  might 
not  be  identical  with  the  arms,  as  they  are  in  the  figure. 


70 


MECHANICS. 


111.  The  Balance. — This  is  a  common  and  valuable  instru- 
ment for  weighing.  It  is  a  straight  lever  with  equal  arms,  having 
scale-pans,  either  suspended  at  the  ends,  or  standing  upon  them, 
one  to  contain  the  poises,  and  the  other  the  substance  to  be 
weighed.  For  scientific  purposes,  particularly  for  chemical  analy- 
sis, great  care  is  bestowed  on  the  construction  of  the  balance. 

The  arms  of  the  balance,  measured  from  the  fulcrum  to  the 
points  of  suspension,  must  be  precisely  equal. 

The  knife-edges  forming  the  fulcrum,  and  the  points  of  sus- 
pension, are  made  of  hardened  steel,  and  arranged  exactly  in  a 
straight  line. 

The  centre  of  gravity  of  the  beam  is  Mow  the  fulcrum,  so  that 
there  may  be  a  stable  equilibrium ;  and  yet  below  it  by  an  exceed- 
ingly small  distance,  in  order  that  the  balance  may  be  very 
sensitive. 

To  preserve  the  edge  of  the  fulcrum  from  injury,  the  beam  is 
raised  by  supports  called  Y's,  when  not  in  use. 

A  long  index  at  right  angles  to  the  beam,  points  to  zero  on  a 
scale  when  the  beam  is  horizontal. 

To  protect  the  instrument  from  dust  and  moisture  at  all  times, 
and  from  air-currents  while  weighing,  the  balance  is  in  a  glass 
case,  whose  front  can  be  raised  or  lowered  at  pleasure. 

FIG.  81. 


A  balance  for  chemical  analysis  is  shown  in  Fig.  81.  By  turn- 
ing the  knob  0,  the  beam  can  be  raised  on  the  Y's  A  A  from  the 
surface  on  which  the  fulcrum  K  rests.  The  screw  C  raises  and 


THE    STEELYARD.  7^ 

lowers  the  fulcrum  in  relation  to  the  centre  of  gravity  of  the  beam, 
in  order  to  increase  or  diminish  the  sensitiveness  of  the  instru- 
ment. In  the  most  carefully  made  balances,  the  index  will  make 
a  perceptible  change,  by  adding  to  the  scale  one  millionth  of  the 
poise. 

For  commercial  purposes,  it  is  convenient  to  have  the  scale- 
pans  above  the  beam.  This  is  done  by  the  use  of  additional  bars, 
which  with  the  beam  form  parallelograms,  whose  upright  sides 
are  rods,  projecting  upward  and  supporting  the  scales.  Such  con- 
trivances necessarily  increase  friction ;  but  balances  so  constructed 
are  sufficiently  sensitive  for  ordinary  weighing. 

112.  The  Steelyard. — This  is  a  weighing  instrument,  having 
a  graduated  arm,  along  which  a  poise  may  be  moved,  in  order  to 
balance  various  weights  on  the  short  arm.  While  the  moment  of 
the  article  weighed  is  changed  by  increasing  or  diminishing  its 
quantity,  that  of  the  poise  is  changed  by  altering  its  acting  dis- 
tance. Since  P  x  A  G  =  W  x  B  C  (Fig.  82),  and  P  is  constant, 

\ 

FIG.  82. 


and  also  the  distance  B  C  constant,  ACccW;  hence,  if  W  is  suc- 
cessively 1  lb.,  2  Ibs.,  3  Ibs.,  &c.,  the  distances  of  the  notches,  «,  b,  c, 
&c.,  are  as  1,  2,  3,  &c. ;  in  other  words,  the  bar  0  D  is  divided  into 
equal  parts.  In  this  case,  the  graduation  begins  from  the  fulcrum 
C  as  the  zero  point. 

But  suppose,  what  is  often  true,  that  the  centre  of  gravity  of 
the  steelyard  is  on  the  long  arm,  and  that  P  placed  at  E  would 
balance  it ;  then  the  moment  of  the  instrument  itself  is  on  the 
side  C  D,  and  equals  P  x  C  K  Hence,  the  equation  becomes 

•P  x  A  C  +  P  x  CE  =  W  x  B  0\  or 
P  x  A  E  =  W  x  B  0. 

:.  TFoc  A  E]  and  the  graduation  must  be  considered  as  com- 
mencing at  E  for  the  zero  point.  Such  a  steelyard  cannot  weigh 
below  a  certain  limit,  corresponding  to  the  first  notch  a. 

To  find  the  length  of  the  divisions  on  the  bar,  divide  A  E,  the 
distance  of  the  poise  from  the  zero  point,  by  W,  the  number  of 
units  balanced  by  P,  when  at  that  distance. 


2%  MECHANICS. 

The  steelyard  often  has  two  fulcrums,  one  for  less  and  the  other 
for  greater  weights. 

113.  Platform  Scales. — This  name  is  given  to  machines 
arranged  for  weighing  heavy  and  bulky  articles  of  merchandise. 
The  largest,  for  cattle,  loaded  wagons,  &c.,  are  constructed  with 
the  platform  at  the  surface  of  the  ground.  In  order  that  the  plat- 
form may  stand  firmly  beneath  its  load,  it  rests  by  four  feet  on  as 
many  levers  of  the  second  order,  whose  arms  have  equal  ratios. 
A  Fy  B  Fy  0  G,  D  G  (Fig.  83),  are  four  such  levers,  resting  on  the 

FIG.  83. 


fulcrums,  A,  B,  (7,  D,  while  the  other  ends  meet  on  the  knife- 
edge,  F  G,  of  another  lever,  L  M.  This  fifth  lever  has  its  fulcrum 
at  L,  and  its  outer  extremity  is  attached  by  a  vertical  rod,  M  N, 
to  a  steelyard,  whose  fulcrum  is  E,  and  poise  P.  The  five  levers 
are  arranged  in  a  square  cavity  just  below  the  surface  of  the 
ground.  The  dotted  line  shows  the  outline  of  the  cavity.  On  the 
bearing  points  of  the  four  levers,  H,  /,  J,  K,  rest  the  feet  of  the 
platform  (not  represented),  which  is  firmly  built  of  plank,  and  just 
fits  into  the  top  of  the  cavity  without  touching  the  sides.  The 
machine  is  a  compound  lever  of  three  parts ;  for  the  four  levers 
act  as  one  at  F  G,  and  are  used  to  give  steadiness  to  the  platform 
which  rests  upon  them. 

A  construction  quite  similar  to  the  above  is  made  of  portable 
size,  and  used  in  all  mercantile  establishments  for  weighing  heavy 
goods. 

114.  Questions  on  the  Lever. — 1.  A  B  (Fig.  84)  is  a  uni- 
form bar,  2  feet  long,  and  weighs 
4  oz. ;  where  must  the  fulcrum  be    B  FlG-  84-      c  . 

put,  that  the  bar  may  be  balanced    _ i 

by  P9  weighing  5  Ibs.  ? 

Ans.  4  of  an  inch  from  A. 

2.  A  lever  of  the  second  order 
is  25  feet  long ;  at  what  distance  from  the  fulcrum  must  a  weight 


QUESTIONS    ON    THE    LEVER.  73 

of  125  pounds  be  placed,  so  that  it  may  be  supported  by  a  power 
able  to  sustain  60  pounds,  acting  at  the  extremity  of  the  lever. 

Ans.  12  feet. 

3.  A  and  B  are  of  the  same  height,  and  sustain  upon  their 
shoulders  a  weight  of  150  pounds,  placed  on  a  pole  9^  feet  long; 
the  weight  is  placed  6§  feet  from  A ;  what  is  the  weight  sustained 
by  each  person  ? 

Ans.  A  sustains  42  f  Ibs.,  and  B  sustains  107 1  Ibs. 

4.  The  longer  arm  of  a  steelyard  is  2  feet  2  inches  in  length, 
and  the  shorter  2f  inches;  and  its  apparatus  of  hooks,  &c.,  is  so 
contrived  that  a  weight  of  2  pounds,  placed  upon  the  longer  arm, 
at  the  distance  of  10  inches  from  the  centre  of  motion,  will  balance 
8  pounds  placed  at  the  extremity  of  the  shorter  arm ;  the  movable 
weight  (of  2  pounds)  cannot  conveniently  be  placed  nearer  to  the 
fulcrum  than  f  of  an  inch ;  what  must  be  the  graduation  of  the 
steelyard  that  it  may  weigh  ounces,  and  what  will  be  the  greatest 
and  least  weights  that  can  be  ascertained  by  it  ? 

Ans.  The  graduation  is  to  12ths  of  an  inch ;  and  it  will 
weigh  from  1  to  20  pounds. 

5.  A  lent  lever,  ACS  (Fig.  85),  has  the  arm  A  G  —  3  feet, 
0  B  =  8  feet,  P  =  5  Ibs.,  and  the  an- 
gle A  C  B  =  140° ;   what  weight,  W, 

must  be  attached  at  J9,  in  order  to 
keep  A  G  horizontal  ? 

Ans.  2.4476  Ibs. 

6.  A  cylindrical  straight  lever  is 
14  feet  long,  and  weighs  6  Ibs.  5  oz. ; 
its  longer  arm  is  9,  and  its  shorter  5 
feet;    at  the  extremity  of  its  shorter 
arm  a  weight  of  15  Ibs.  2  oz.  is  sus- 
pended ;  what  weight  must  be  placed 
at  the  extremity  of  the  longer  arm  to 

keep  it  in  equilibrium  ?  Ans.  7  Ibs. 

7.  A  uniform  bar,  12  feet  long,  weighs  7  Ibs. ;  a  weight  of  10 
Ibs.  hangs  on  one  end,  and  2  feet  from  it  is  applied  an  upward 
force  of  25  Ibs. ;  where  must  the  fulcrum  be  put  to  produce  equi- 
librium? Ans.  1  foot  from  the  10  Ibs. 

8.  The  lengths  of  the  arms  of  a  balance  are  a  and  b.    "When  p 
ounces  are  hung  on  a,  they  balance  a  certain  body ;  but  it  requires 
q  ounces  to  balance  the  same  body,  when  placed  in  the  other  scale. 
What  is  the  true  weight  of  the  body  ?    According  to  the  first 
weighing,  ap  —  ~bx\  according  to  the  second,  ~b  q  =  a x.    :.a~bpq 
=  a  b  y\  and  x  =  VjTq.    Hence,  the  true  weight  is  a  geometrical 
mean  between  the  apparent  weights. 


74- 


MECHANICS. 


9.  On  one  arm  of  a  false  balance  a  body  weighs  11  Ibs. ;  on  the 
other,  17  Ibs.  3  oz.;  what  is  the  true  weight? 

Am.  13  Ibs.  12  oz. 

10.  Four  weights  of  1,  3,  5,  7  Ibs.,  respectively,  are  suspended 
from  points  of  a  straight  lever,  eight  inches  apart;  how  far  from 
the  point  of  suspension  of  the  first  weight  must  the  fulcrum  be 
placed,  that  the  weights  may  be  in  equilibrium  ? 

Ans.  17  inches. 

11.  Two  weights  keep  a  horizontal  lever  at  rest,  the  pressure 
on  the  fulcrum  being  10  Ibs.,  the  difference  of  the  weights  4  Ibs., 
and  the  difference  of  the  lever  arms  9  inches;  what  are  the  weights 
and  their  lever  arms  ? 

Ans.  Weights,  7  Ibs.  and  3  Ibs. ;  arms,  6|  in.  and  15|  in. 


FIG.  86. 


II.  THE  WHEEL  AKD  AXLE. 

115.  Description  and  Law  of  the  Machine. — The  wheel 
and  axle  consists  of  a  cylinder  and  a  wheel,  firmly  united,  and  free 
to  revolve  on  a  common  axis.  The  power  acts  at  the  circumfer- 
ence of  the  wheel  in  the  direction  of  a  tangent,  and  the  weight  in 
the  same  manner,  at  the  circumference  of  the  cylinder  or  axle ;  so 
that  the  acting  distances  are  the  radii  at  the  two  points  of  contact. 
As  the  system  revolves,  the  radii  successively  take  the  place  of 
acting  distances,  without  altering  at  all  the  relation  of  the  forces 
to  each  other.  The  wheel  and  axle  is  therefore  a  kind  of  endless 
lever. 

Let  W  (Fig.  86)  be  the  weight  suspended  from  the  axle,  tend- 
ing to  revolve  it  on  the  line  L  M\ 
and  P,  the  power  acting  on  the 
wheel,  tending  to  revolve  the  sys- 
tem in  the  opposite  direction.  It 
is  plain  that  the  acting  distances 
are  the  radius  of  the  axle,  and  A  C 
the  radius  of  the  wheel.  In  case  of 
equilibrium,  the  moment  of  W 
equals  the  moment  of  P.  Calling 
the  radius  of  the  axle  r,  and  the 
radius  of  the  wheel  R,  then  W  x  r 
=  P  x  R-,  or 

P\  WiinR. 

If,  instead  of  the  weight  P,  suspended  on  the  wheel,  the  rope 
be  drawn  by  any  force  in  the  direction  P'  or  P",  it  is  still  tangent 
to  the  circumference,  and  therefore  its  acting  distance,  C  Dor  C  B, 
the  same  as  before.  In  general,  the  law  of  equilibrium  for  this 
machine  is, 


THE    WHEEL    AND    AXLE. 


75 


FIG.  87. 


The  power  is  to  the  weight  as  the  radius  of  the  axle  to  the  radius 
of  the  wheel. 

If  the  rope  on  the  wheel,  being  fastened  at  A  (Fig.  87)  is 
drawn  by  the  side  of  the  wheel,  as  A  Pf, 
the  acting  distance  of  the  power  is  dimin- 
ished from  G  A  to  G  E,  and  therefore  its 
efficiency  is  diminished  in  the  same  ratio. 
Were  the  rope  drawn  away  from  the 
wheel,  as  A  P",  making  an  equal  angle 
on  the  other  side  of  A  P,  the  same  effect 
is  produced,  the  acting  distance  now  be- 
coming C  F. 

The  radius  of  the  wheel  and  the  radius 
of  the  axle  should  each  be  reckoned  from 
the  axis  of  rotation  to  the  centre  of  the 

rope;  that  is,  half  of  the  thickness  of  the  rope  should  be  added  to 
the  radius  of  the  circle  on  which  it  is  coiled.  Calling  t  the  half 
thickness  of  the  rope  on  the  axle,  and  t'  that  of  the  rope  on  the 
wheel,  the  proportion  for  equilibrium  is  P  :  W  :  :  r  +  t  : 
E  +  t'. 

116.  The  Compound  Wheel  and  Axle.  —  When  a  train 
of  wheels,  like  that  in  Fig.  88,  is  put  in 
motion,  those  which  communicate  mo- 
tion  by  the  circumference  are  called 
driving  wheels,  as  A  and  G;  those  which 
receive  motion  by  the  circumference  are 
called  driven  wheels.  And  the  law  of 
equilibrium  is, 

The  power  is  to  the  weight  as  the  pro- 
duct of  the  radii  of  the  driving  wheels 
to  the  product  of  the  radii  of  the  driven 
wheels. 

The  crank  P  Q  is  to  be  reckoned 
among  driven  wheels  ;  the  axle  E  among  driving  wheels. 

Let  the  radius  of  B  be  called  fi;  of  D,  R  ';  of  A,  r\  of  <7,  r'  ; 
of  E,  r".  Call  the  force  exerted  by  A  on  B,  x  ;  that  of  C  on  D,  y. 
Then 

P:x  ::r   :  P  Q; 
x:y.:r':R; 


FlG- 


/.  P:  W:  :  r  x  r'  x  r"  :  P  Q  x  R  x  R'. 
If  the  driving  wheels  are  equal  to  each  other,  and  also  the 
driven  wheels,  and  the  number  of  each  is  n,  then 
P:  W::rn:  Rn. 


76  MECHANICS. 

117.  Direction   and   Rate    of  Revolution. — When  two 
wheels  are  geared  together  by  teeth,  they  necessarily  revolve  in 
contrary  directions.    Hence,  in  a  train  of  wheels,  the  alternate 
axles  revolve  the  same  way. 

The  circumferences  of  two  wheels  which  are  in  gear  move  with 
the  same  velocity ;  hence  the  number  of  revolutions  will  be  re- 
ciprocally as  the  radii  of  the  wheels. 

Since  teeth  which  gear  together  are  of  the  same  size,  the  rela- 
tive number  of  teeth  is  a  measure  of  the  relative  circumferences, 
and  therefore  of  the  relative  radii  of  the  wheels.  If  the  wheel  A 
(Fig.  88)  has  20  teeth,  and  B  has  40,  and  again  if  C  has  15,  and 
D  45,  then  for  every  revolution  of  B,  A  revolves  twice,  and  for 
every  revolution  of  D,  C  revolves  three  times.  Therefore,  six 
turns  of  the  crank  are  necessary  to  give  one  revolution  to  the 
axle  E. 

By  cutting  the  teeth  of  wheels  on  a  conical  instead  of  a  cylin- 
drical surface,  the  axles  may  be  placed  at  any  angle  with  each 
other,  as  represented  in  Fig.  89. 

Whether  axles  are  parallel  or  not,  lands  in- 
stead  of  teeth  may  be  used  for  transmitting 
rotary  motion.  But  as  bands  are  liable  to  slip 
more  or  less,  they  cannot  be  employed  in  cases 
requiring  exact  relations  of  velocity. 

118.  Questions  on   the  Wheel    and 
Axle.— 

1.  A  power  of  12  Ibs.  balances  a  weight  of 
100  Ibs.  by  a  wheel  and  axle;  the  radius  of  the 
axle  is  6  inches ;  what  is  the  diameter  of  the  wheel  ? 

Am.  8  ft.  4  in. 

2.  W=  500  Ibs.;  R  =  4  ft;  r  =  8  in.;  the  weight  hangs  bya 
rope  1  inch  thick,  but  the  power  acts  at  the  circumference  of  the 
wheel  without  a  rope ;  what  power  will  sustain  the  weight  ? 

Ans.  88.54  Ibs. 

3.  R  =  1  ft;  r  =  2  in.;  the  well-stone  weighs  256  Ibs.;  the 
bucket,  empty,  weighs  18  Ibs. ;  the  bucket,  filled,  weighs  65  Ibs. ; 
what  force  must  a  person  apply  to  the  bucket-rope,  in  each  case, 
for  equilibrium?  Ans.  1st,  down,  24|  Ibs.;  2d,  up,  22  J  Ibs. 

4.  In  Fig.  88,  A  and  C  have  each  15  teeth,  B  and  D  each  40 
teeth ;  the  radius  of  the  axle  E  is  4  inches ;  the  rope  on  it  1  inch 
in  diameter ;  and  the  radius  of  the  crank  P  Q  is  18  inches ;  what 
is  the  ratio  of  power  to  weight  in  equilibrium?        Ans.  1 :  28$. 


THE    PULLEY. 


77 


III.  THE  PULLET. 

119.  The  Pulley  Described. — The  pulley  consists  of  one  or 
more  wheels  or  rollers,  with  a  rope  passing  over  the  edge  in  which 
a  groove  is  sunk  to  keep  the  rope  in  place.     The  axis  of  the  roller 
is  in  a  block,  which  is  sometimes  fixed,  and  sometimes  rises  and 
falls  with  the  weight;  and  the  pulley  is  accordingly  called  a  fixed 
pulley  or  a  movable  pulley.    The  principle  which  explains  the 
relation  of  power  and  weight  in  every  form  of  pulley,  is  this : 

Whatever  strain  or  tension  is  applied  to  one  end  of  a  cord,  is 
transmitted  through  its  whole  length,  if  it  does  not  branch,  however 
much  its  direction  is  changed. 

In  the  pulley,  the  sustaining  portions  of  the  rope  are  assumed 
to  be  parallel  to  each  other. 

120.  The  Fixed  Pulley. — In          FIG.  90.  FIG.  91. 

the  fixed  pulley,  A  (Fig.  90),  the 
force  P  produces  a  tension  in  the 
string,  which  is  transmitted  through 
its  whole  length,  and  which  can  be 
balanced  only  when  W  equals  P. 
Hence,  in  the  fixed  pulley,  the 
poiver  and  weight  are  equal.  This 
machine  is  useful  for  changing  the 
direction  in  which  the  force  is  ap- 
plied to  the  weight ;  and  if  the 
power  only  acts  in  the  plane  of  the 
groove  of  the  wheel,  it  is  immaterial  what  is  its  direction,  horizon- 
tal, vertical,  or  oblique. 

121.  The  Movable  Pulley.— In  Fig.  91,  the  ten-     FIG.  92. 
sion  produced  by  P,  is  transmitted  from  A  down  to 

the  wheel  E,  and  thence  up  to  D ;  therefore  W  is  sus- 
tained by  two  portions  of  the  rope,  each  of  which 
exerts  a  force  equal  to  P.. 

.-.  W=2P',  or  P:  TF::1:2. 

The  same  reasoning  applies,  where  the  rope  passes 
between  the  upper  and  lower  blocks  any  number  of 
times,  as  in  Fig.  92.  The  force  causes  a  tension  in  the 
rope,  which  is  transmitted  to  every  portion  of  it.  If  n 
is  the  number  of  portions  which  sustain  the  lower 
block,  then  W  is  upheld  by  n  P ;  and  if  there  is  equi- 
librium, P  :  W : :  1  :  n.  In  the  figure,  the  weight 
equals  six  times  the  power.  The  law  of  equilibrium, 
therefore,  for  the  movable  pulley  with  one  rope,  is  this, 

Tlie  power  is  to  the  iveight  as  one  to  the  number  of 


rVQ 
10 


MECHANICS. 


the  sustaining  portions  of  the  rope;  or,  as  one  to  twice  the  number 
of  movable  pulleys. 

122.  The  Compound  Pulley.  —  Wherever  a  system  of  pul* 
leys  has  separate  ropes,  the  machine  is  to  be  regarded  as  com- 
pound, and  its  efficiency 

is  calculated  according-  FlG-  93-  F10-  94- 

ly.  Figures  93  and  94 
are  examples.  In  Fig. 
93,  call  the  weight  sus- 
tained by  F}  x,  and  that 
sustained  by  D,  y.  Then 
(Art.  131), 

P:  «::1:2; 

x:      ::l:2 


Vv 


And  if  n  is  the  number 
of  ropes, 

P  :  W  :  :  1  :  2". 

In  Fig.  94,  the  tension  P  is  transmitted 
over  A  directly  to  the  weight  at  G  ;  the  wheel 
A  is  loaded,  therefore,  with  2  P,  and  a  tension 
of  2  P  comes  upon  the  second  rope,  which  is 
transmitted  over  B  to  the  weight  at  F.    In 
like  manner,  a  tension  of  4  P  is  transmitted 
over  G  to  K    The  sum  of  all  these  being  ap- 
plied to  the  weight,  it  must  therefore  be  equal 
to  that  sum  in  case  of  equilibrium.    Therefore,  P  :  W  :  :  1  :  1  + 
2  +  4  +  &c.    Now  the  sum  of  this  geometrical  series  to  n  terms 
is  2rt  —  1  ;  /.  P  :  W:  :  1  :  Zn  —  1.    This  combination  is  therefore  a 
little  less  efficient  than  the  preceding. 

Since  the  several  ropes  have  different  tensions,  the  weight  can- 
not be  balanced  upon  them,  unless  those  of  greatest  tension  are 
nearest  the  line  of  direction  of  the  body.  For  example,  if  the  rope 
F  is  directed  toward  the  centre  of  gravity  of  the.  weight,  the  rope 
G  should  be  attached  four  times  as  far  from  it  as  the  rope  E9  in 
order  to  prevent  the  weight  from  tipping. 

The  pulley  owes  its  efficiency  as  a  machine  to  the  fact,  that  the 
tension  produced  by  the  power  is  applied  repeatedly  to  the  weight. 
The  only  use  of  the  wheels  is  to  diminish  friction.  Were  it  not 
for  friction,  the  rope  might  pass  round  fixed  pins  in  the  blocks, 
and  the  ratio  of  power  to  weight  would  still  be  in  every  case  the 
same  as  has  been  shown. 


THE    ROPE    MACHINE. 


79 


FIG.  95. 


IV.  THE  ROPE  MACHINE. 

123.  Definition  and  Law  of  this  Machine. — 

The  rope  machine,  is  one  in  which  the  power  and  weight  are  in 
equilibrium  by  the  tension  of  one  or  more  ropes. 

According  to  this  definition  the  pulley  is  included.  It  is  that 
particular  form  of  the  rope  machine  in  which  the  sustaining  parts 
of  the  ropes  are  parallel ;  and  it  is  treated  as  a  separate  machine, 
because  its  theory  is  very  simple,  and  because  it  is  used  far  more 
extensively  than  any  other  forms. 

If  the  two  portions  of  rope 
which  sustain  the  weight  are  in- 
clined,  as  in  Fig.  95,  then  W  is 
no  longer  equal  to  the  sum  of 
their  tensions,  as  it  is  in  the  pul- 
ley, but  is  always  less  than  that, 
according  to  the  following  law : 

The  power  is  to  the  weight  as 
radius  is  to  twice  the  cosine  of 
half  the  angle  between  the  parts 
of  the  rope. 

Put    A  E  B  =  2  a\    then 
FED  =  a,  and  since  sin  B  E  W 
=  sin  B  ED  =  sin  a,  we  shall  have  (Art.  61)  P  :  W:  •  sin  a  :  sin 
2  a ;  but  sin  a  :  sin  2  a  : :  R  :  2  cos  a ;  .*.  P  :  W : :  R  :  2  cos  a. 

If  in  Fig.  96,  the  end  of  the  cord,  instead  of  being  attached  to 
the  beam,  is  carried  over  another  fixed  pulley,  and  a  weight  equal 
to  P  is  hung  upon  it,  the  equilibrium  will  be  preserved,  because  all 
parts  of  the  rope  have  a  tension  equal  to  P\  therefore,  as  before, 

P:  TF::  72:2  cos  «;  or  2  P  :  TF::72:cos«  \\BC\  CD. 

124.  Change  in  the  Ratio  of  Power  and  Weight— If 

P  is  given,  all  the  possible  values  of  W  are  included  between 
JF=0,  and  W=2P. 

When  the  rope  is  straight  from  A  to  J5,  so  that  C  D  —  0,  then, 
by  the  above  proportion,  W  =  0.  As  W  is  increased  from  zero, 
the  point  0  descends;  and 
when  D  0  =  J  B  C,  then,  by 
the  proportion,  W=  P.  In 
that  case  D  0  B  =  60°,  and 
the  angles,  A  C  B,  A  C  W, 


and  B  C  W,  are  equal  (each 
being  120°),  as  they  should  be, 
because  each  of  the  equal 
forces,  P,  P,  and  JF,  is  as  the 
sine  of  the  angle  between  the  directions  of  the  other  two. 


80 


MECHANICS. 


But  when  W  has  increased  to  2  P,  it  descends  to  an  infinite 
distance;  for  then,  by  the  proportion,  CD  =  B  C,  that  is,  the  side 
of  a  right-angled  triangle  is  equal,  to  the  hypothenuse.  Thus,  the 
extreme  values  of  W  are  0  and  2  P. 

It  appears  from  the  foregoing,  that  a  perfectly  flexible  rope, 
having  weight  cannot  be  drawn  into  a  straight  horizontal  line,  by 
any  force  however  great ;  for  C  cannot  coincide  with  Z>,  except 
when  W  —  0. 

125.  The  Branching  Rope.— When  C9  where  the  weight  is 
suspended,  is  a  fixed  point  of  the  rope, 
we  have  a  branching  rope,  and  the 
principle  of  transmitted  tension  does 
not  apply  beyond  the  point  of  division. 

Let  P,  >  and  W  (Fig.  97),  be 
given,  and  C  a  fixed  point  of  the  rope. 
Produce  W  C,  and  let  A  E,  drawn 
parallel  to  G  B,  intersect  it  in  K  The 
sides  of  A  G  E  are  proportional  to  the 
given  forces ;  therefore  its  angles  can 
be  found,  and  the  inclinations  of  A  G 
and  B  G  to  the  vertical  G  W  are  known. 

128.  The  Funicular  Polygon.— If  several  weights  are  at- 
tached at  fixed  points  along  the  cord  A  G  B  (Fig.  98),  the  combi- 
nation is  called  the  funicular  polygon ; 
and  the  fact  that  there  are  opposite 
and  equal  tensions  in  any  portion  of 
the  cord  will  enable  us  to  transfer  all 
the  forces  to  one  point. 

Let  the  tension  of  G  D  =  T\  and 
that  of  D  E  =  T'.  C  is  kept  at  rest 
by  P,  T,  and  W\  hence  T  is  equal  to 

the  resultant  of  P  and  W.  But  the  same  T  (in  the  opposite  di- 
rection) equals  and  balances  the  resultant  of  T'  and  W '.  Suppose, 
now,  C  D  to  vanish,  by  removing  G  to  D ;  draw  A  D  parallel  to 
A  G\  let  P  act  in  the  line  D  A,  and  Win  the  line  D  W.  Dwill 
now  be  in  equilibrium,  as  before,  because  there  has  merely  been  made 
a  substitution  of  P  and  W  in  their  original  directions  for  T,  their 
equivalent.  Now  consider  D  to  be  acted  on  by  three  forces,  T',  P, 
and  W  +  W1',  .:  T'  =  the  resultant  of  P  and  W  +  W,  and  the 
two  latter  can  be  transferred,  as  before,  to'  E,AE  being  parallel  to 
A  G  or  A  D.  E  is  therefore  kept  at  rest  by  the  three  forces,  P,  P', 
and  W  -\-  W'  +  W".  We  can  now  use  the  triangle  of  forces,  as  in 
the  preceding  article,  to  determine  the  directions  of  B  E  and  A  E, 
or  its  parallel,  A  C,  and  hence,  of  the  parts,  C  D  and  D  E, 


FIG.  98. 


THE    INCLINED    PLANE. 


81 


If  the  weights  are  all  equal,  and  their  number  =  n,  the  three 
forces  at  E  are  P,  F',  and  n  W. 

An  example  of  this  kind  occurs  in 
the  suspension  bridge,  whose  weight  is 
distributed  at  equal  distances  along 
the  supporting  chains.  And  an  ex- 
treme case  is  that  of  a  heavy  rope  or 
chain  suspended  loosely  over  pulleys, 
as  in  Fig.  99.  Equal  weights  are  sus- 
pended at  an  infinite  number  of  points, 
and  therefore  the  funicoilar  polygon  becomes  a  curve,  and  is  called 
the  catenary  curve.  Its  directions  at  the  extremities,  A  and  B, 
and  the  law  of  the  curve,  may  be  determined  by  the  principles 
given  above. 

V.  THE  INCLINED  PLAICE. 

127.  Relation  of  Power,  Weight,  and  Pressure  on  the 
Plane. — The  mechanical  efficiency  of  the  inclined  plane  is  ex- 
plained on  the  principle  of  oblique  action  ;  that  is,  it  enables  us  to 
apply  the  power  to  balance  or  overcome  only  one  component  of  the 
weight,  instead  of  the  whole.  Let  the  weight  of  the  body  G,  lying 
on  the  inclined  plane  A  0  (Fig.  100),  be  represented  by  W\  and 
resolve  it  into  F  parallel,  and  N  perpendicular  to  the  plane.  N 
represents  the  perpendicular  pressure,  and  is  equal  to  the  reaction 
of  the  plane ;  F  is  the  force  by  which  the  body  tends  to  move 
down  the  plane. 

Let  a  —  the  angle  C,  the  inclination  of  the  plane ;  therefore 
WG  JV  =  a.  Then  F  —  W .  sin  a;  and  N=  W  .cos  a. 


FIG.  100. 


FIG.  101. 


Now  suppose  a  force  P  is  applied  at  0  (Fig.  101),  which  keeps 
the  body  at  rest.  Then  the  resultant  of  Wand.  P  must  be  N, 
which  is  resisted  by  the  plane ;  therefore,  .  y  (. 

P  :  W: :  sin  G  N  P,  or  sin  a  :  sin  P  G  N. 
When  the  power  acts  parallel  to  the  plane,  P  G  N  =  90°,  and 
we  have  P  :  W : :  sin  a  :  sin  90°  : :  A  B  :  A  C.    Hence,  when  the 
6 


82'  MECHANICS. 

power  acts  in  a  line  parallel  to  the  inclined  plane,  which  is  the 
most  common  direction, 

The  power  is  to  the  weight  as  the  height  to  the  length  of  the  in- 
clined plane. 

When  the  power  acts  in  a  line  parallel  to  the  base  of  the  in- 
clined plane,  P  G  N  =  90°  —  #,  and  we  have  P  :  W  : :  sin  a  : 
cos  a  : :  A  B  :  B  C.  Hence,  when  the  power  acts  in  a  line  parallel 
to  the  base  of  the  inclined  plane, 

The  power  is  to  the  weight  as  the  height  is  to  the  base  of  the  in- 
clined plane. 

128.  Power  most  Efficient  when  Acting  Parallel  to 
the  Plane. — From  the  proportion 

P  :  W : :  sin  a  :  sin  P  G  N,  we  derive 
TF=P.sinP  GN 

sin  a 

Now  as  P  and  sin  a  are  given,  W  varies  as  sin  P  G  N,  which 
is  the  greatest  possible  when  P  G  N  =  90° ;  that  is,  when  the 
power  acts  in  a  line  parallel  to  the  plane. 

Whether  the  angle  P  G  N  diminishes  or  increases  from  90°, 
its  sine  diminishes,  and  becomes  zero,  when  P  G  JV=  0°,  or  180°. 
Therefore  W  —  0,  or  no  weight  can  be  sustained,  when  the  power 
acts  in  the  line  G  N,  perpendicular  to  the  plane,  either  toward  the 
plane  or  from  it. 

129.  Expression   for  Perpendicular  Pressure. — From 
the  triangle  P  G  J^we  obtain 

N-.  W:  :  sin  #  P  N:  sin  P  G  N, 
or  N:  W-. :  sin  P  G  W :  sin  P  G  N; 
.  y=  W  sm^JT. 

•   •    U.V     — —      if  •  -,-fc      ^v      -»-r 

sin  P  G  N 
If  the  power  acts  in  a  line  parallel  to  the  inclined  plane, 

P  GW  =90°  +«,P^JV=90°,and^=  W  ^J^t  ^  = 

W  cos  a. 

If  the  power  acts  in  a  line  parallel  to  the  base  of  the  inclined 

plane,  P  G  W  =  90°,  P  G  N  =  90°  -  a,  and  N  =  W  -—  = 

cos  a 

W  sec  a. 

If  the  power  acts  in  a  line  perpendicular  to  the  inclined  plane, 

P  G  W  =  a,  P  G  N  =  0°,  and  N  =  TF-~  =  «. 

130.  Equilibrium  between  Two  Inclined  Planes. — If  a 

body  rests,  as  represented  in  Fig.  102,  between  two  inclined  planes, 


THE    INCLINED    PLANE. 


83 


FIG.  103. 


the  three  forces  which  retain  it  are  its  weight,  and  the  resistances 
of  the  planes.  Draw  H  F  and  L  F 
perpendicular  to  the  planes  through 
the  points  of  contact,  and  G  F  verti- 
cally through  the  centre  of  gravity  of 
the  body.  Since  the  body  is  in  equi- 
librium, these  three  lines  will  pass 
through  the  same  point  (Art.  60,  2). 
Let  that  point  be  F,  and  draw  G  P 
parallel  to  L  F9  and  M  K  parallel  to 
the  horizon.  GPFis  similar  to  K  C  M. 
Therefore  (since  Pressure  on  A  C :  Pr. 
onD  Q-.'.P  G-.FP}, 


Pressure  on  A  G :  Pr.  on  D  C 


KC-.MC, 

sin  M :  sin  K, 

sinD  C'EismA  C  B. 


FIG.  103. 
A 


That  is,  when  a  body  rests  between  two  planes,  it  exerts  pressures 
on  them  which  are  inversely  as  the  sines  of  their  inclinations  to 
the  horizon. 

If,  therefore,  one  of  the  planes  is  horizontal,  none  of  the 
pressure  can  be  exerted  on  any  other  plane.  It  is  friction  alone 
which  renders  it  possible  for  a  body  on  a  horizontal  surface  to 
lean  against  a  vertical  wall. 

131.  Bodies  Balanced  on  Two   Planes  by  a  Cord 
passing  over  the  Ridge,— Let  P  and  W  balance  each  other  on 
the  planes  A  D  and  A  C  (Fig.  103), 

which  have  the  common  height  A  B, 
by  means  of  a  cord  passing  over  the 
fixed  pulley  A.  The  tension  of  the 
cord  is  the  common  power  which  pre- 
vents each  body  from  descending ;  and 
as  the  cord  is  parallel  to  each  plane, 
we  have  (calling  the  tension  t), 

t'.P  '.\AB:AD', 

and    t :  W  : :  A  B  :  A  <7; 

/.    P  \W\\AD\AC\ 

that  is,  the  weights,  in  case  of  equilibrium,  are  directly  as  the 
lengths  of  the  planes. 

132.  Questions  on  the  Inclined  Plane. — 

1.  If  a  horse  is  able  to  raise  a  weight  of  440  Ibs.  perpendicu- 
larly, what  weight  can  he  raise  on  a  railway  having  a  slope  of  five 
degrees?  Ans.  5048.5  Ibs. 


84  MECHANICS. 

2.  The  grade  of  a  railroad  is  20  feet  in  a  mile ;  what  power 
must  be  exerted  to  sustain  any  given  weight  upon  it  ? 

Ans.  1  Ib.  for  every  264  Ibs. 

3.  What  force  is  requisite  to  hold  a  body  on  an  inclined  plane, 
by  pressing  perpendicularly  against  the  plane  ? 

Ans.  An  infinite  force. 

4.  A  certain  power  was  able  to  sustain  500  tons  on  a  plane  of 
7i° ;  but  on  another  plane,  it  could  sustain  only  400  tons ;  what 
was  the  inclination  of  the  latter  ?  Ans.  9°  23'  25". 

5.  Equilibrium  on  an  inclined  plane  is  produced  when  the 
power,  weight,  and  perpendicular  pressure  are,  respectively,  9, 13, 
and  6  Ibs. ;  what  is  the  inclination  of  the  plane,  and  what  angle 
does  the  power  make  with  the  plane  ? 

Ans.  a  -  37°  21'  26".    Inclination  of  power  to  plane 
=  28°  46'  54". 

6.  A  power  of  10  Ibs.,  acting  parallel  to  the  plane,  supports  a 
certain  weight ;  but  it  requires  a  power  of  12  Ibs.  parallel  to  the 
base  to  support  it.    What  is  the  weight  of  the  body,  and  what  is 
the  inclination  of  the  plane  ? 

Ans.  W=  18.09  Ibs.    a  =  33°  33'  25" 

7.  To  support  a  weight  of  500  Ibs.  upon  an  inclined  plane  of 
50°  inclination  to  the  horizon,  a  force  is  applied  whose  direction 
makes  an  angle  of  75°  with  the  horizon.    What  is  the  magnitude 
of  this  force,  and  the  pressure  of  the  weight  against  the  plane  ? 

Ans.  P  =  422.6  Ibs.    N  =  142.8  Ibs. 

VL  THE  SCEEW. 

133.  Reducible  to  the  Inclined  Plane.— The  screw  is  a 
cylinder  having  a  spiral  ridge  or  thread  around  it,  which  cuts  at  a 
constant  oblique  angle  all  the  lines  of  the  surface  parallel  to  the 
axis  of  the  cylinder.  A  hollow  cylinder,  called  a  nut,  having  a 
similar  spiral  within  it,  is  fitted  to  move  freely  upon  the'  thread  of 
the  solid  cylinder.  In  Fig.  104,  let  the  base  A  B  of  the  inclined 

FIG.  104. 


THE    SCREW    AND    LEVER. 


85 


plane  A  C  be  equal  to  twice  the  circumference  of  the  cylinder 
A'  E\  then  let  the  plane  be  wrapped  about  the  cylinder,  bringing 
the  points  A,  F,  and  B,  to  the  point  A':,  then  will  A  C  describe 
two  revolutions  of  the  thread  from  A'  to  C'.  Therefore  the  me- 
chanical relations  of  the  screw  are  the  same  as  of  the  inclined 
plane. 

If  a  weight  be  laid  on  the  thread  of  the  screw,  and  a  force  be 
applied  to  it  horizontally  in  the  direction  of  a  tangent  to  the 
cylinder,  the  case  is  exactly  analogous  to  that  of  a  body  moved  on 
an  inclined  plane  by  a  force  parallel  to  the  base.  Let  r  be  the 
radius  of  the  cylinder,  then  2  rr  r  is  the  circumference  ;  also  let  d 
be  the  distance  between  the  threads,  (that  is,  from  any  point  of 
one  revolution  to  the  corresponding  point  of  the  next,)  measured 
parallel  to  the  axis  of  the  cylinder ;  then  2  n  r  is  the  base  of  an 
inclined  plane,  and  d  its  height.  Therefore  (Art.  127), 
P:  WiidiZKr;  or, 

The  power  is  to  tlie  weight  as  the  distance  between  the  threads 
measured  parallel  to  the  axis,  is  to  the  circumference  of  the  screw. 

If  instead  of  moving  the  weight  on  the  thread  of  the  screw,  the 
force  is  employed  to  turn  the  screw  itself,  while  the  weight  is  free 
to  move  in  a  vertical  direction,  the  law  is  the  same.  Thus, 
whether  the  screw  A'  E  is  allowed  to  rise  and  fall  in  the  fixed  nut 
G  H,  or  whether  the  nut  rises  and  falls  on  the  thread  of  the  screw, 
while  the  latter  is  revolved,  without  moving  longitudinally,  in 
each  case,  P  :  W : :  d :  2  TT  r. 

134.  The  Screw  and  Lever  Combined. — The  screw  is  so 
generally  combined  with  the  lever  in  practical  mechanics,  that  it 
is  important  to  present  the  law  of  the 
compound  machine.  Let  A  F  (Fig.  105) 
be  the  section  of  a  screw,  and  suppose 
B  C,  a  lever  of  the  second  order,  to  be 
applied  to  turn  it.  The  fulcrum  is  at  C, 
the  power  acts  at  B,  and  the  effect  pro- 
duced by  the  lever  is  at  A,  the  surface  of 
the  cylinder.  Call  that  effect  x,  and  let 
d  =  the  distance  between  the  threads ; 
then, 

P:    xi:AC:BC, 
and    x:W::      d:2n  A  C; 
compounding  and  reducing,  we  have 

P:  W::      d\%KBC\  that  is, 

The  power  is  to  the  iveight  as  the  distance  between  the  threads, 
measured  parallel  to  the  axis,  to  the  circumference  described  by  the 
power. 


FIG.  105. 


86  MECHANICS. 

The  law  as  thus  stated  is  applicable  to  the  screw  when  used 
with  the  lever  or  without  it. 

135.  The  Endless  Screw. — The  screw  is  so  called,  when  its 
thread  moves  between  the  teeth  of  a  wheel,  thus  causing  it  to 
revolve.    It  is  much  used  for  diminish- 
ing very  greatly  the  velocity  of  the  FlG- 106- 
weight. 

Let  P  Q  (Fig.  106)  be  the  radius  of 
the  crank  to  which  the  power  is  ap- 
plied; d,  the  distance  between  the 
threads;  R,  the  radius  of  the  wheel; 
r,  the  radius  of  the  axle ;  and  call  the 
force  exerted  by  the  thread  upon  the 
teeth,  x.  Then, 

P:    x::  d  :  2  TT  x  P  Q, 
end  x-.Wi:  r  :  R; 
.:  P:  W::dr:%7T  x  R  x  P  Q. 

If,  for  example,  P  Q  —  30  inches,  d  —  1  in.,  R  =  18  in. ; 

r  +  t  —  2  in. ;  then  W  moves  with  1696  times  less  velocity  than  P. 
f-.ia.^'u^  • 

136.  The  Right  and  Left  Hand  Screw.— The  common 
form  of  screw  is  called  the  right-hand  screw,  and  may  be  described 
thus ;  if  the  thread  in  its  progress  along  the  length  of  the  cylinder, 
passes  from  the  left  over  to  the  right,  it  is  called  a  right-hand  screw. 
Hence,  a  person  in  driving  a  screw  forward  turns  it  from  his  left 
over  (not  under)  to  his  right,  and  in  drawing  it  back  he  reverses 
this  movement.    Fig.  104  represents  a  right-hand  screw. 

The  left-hand  screw  is  one  whose  thread  is  coiled  in  the  oppo- 
site direction, — that  is,  it  advances  by  passing  from  right  over  to 
left.  This  kind  is  used  only  when  there  is  special  reason  for  it. 
For  example,  the  screws  which  are  cut  upon  the  left-hand  ends  of. 
carriage  axles  are  left-hand  screws;  otherwise  there  would  be 
danger  that  the  friction  of  the  hub  against  the  nut  might  turn 
the  nut  off  from  the  axle.  Also,  when  two  pipes  for  conveying 
gas  or  steam  are  to  be  drawn  together  by  a  nut,  one  must  have  a 
right-hand,  and  the  other  a  left-hand  screw. 

137.  Questions  on  the  Screw. — 

1.  The  distance  between  the  threads  of  a  screw  is  one  inch,  the 
bar  is  two  feet  long  from  the  axis,  and  the  power  is  30  Ibs. ;  what 
is  the  weight  or  pressure  ?  Ans.  4523.76  Ibs. 

2.  The  bar  is  three  feet  long,  reckoned  from  the  axis,  P  — 
60  Ibs.,  W  =  2240  Ibs. ;  -what  is  the  distance  between  the  threads  ? 

Ans.  6.058  inches. 


THE    WEDGE. 


87 


3.  A  compound  machine  consists  of  a  crank,  an  endless  screw, 
a  wheel  and  axle,  a  pulley,  and  an  inclined  plane.  The  radius  of 
the  crank  is  18  inches ;  the  distance  between  the  threads  of  the 
screw,  one  inch  ;  the  radius  of  the  wheel  on  which  the  screw  acts, 
two  feet ;  the  radius  of  the  axle,  6  inches ;  the  pulley  block  has  two 
movable  pulleys  with  one  rope ;  and  the  inclination  of  the  plane  to 
the  horizon  is  30°.  What  weight  on  the  plane  will  be  balanced  by 
a  power  of  100  Ibs.  applied  to  the  crank  ?  Ans.  361911.168  Ibs. 

VII.  THE  WEDGE. 

138.  Definition   of   the   Wedge,    and   the   Mode   of 

Using. — The  usual  form  of  the  wedge  is  a  triangular  prism,  two 
of  whose  sides  meet  at  a  very  acute  angle.  This  machine  is  used 
to  raise  a  weight  by  being  driven  as  an  inclined  plane  underneath 
it,  or  to  separate  .the  parts  of  a  body  by  being  driven  between 
them.  When  it  is  used  by  itself,  and  does  not  form  part  of  a 
compound  machine,  force  is  usually  applied  by  a  blow,  which  pro- 
duces an  intense  pressure  for  a  short  time,  sufficient  to  overcome 
a  great  resistance. 

139.  Law  of  Equilibrium. — Whatever  be  the  direction  of 
the  blow  or  force,  we  may  suppose  it  to  be  resolved  into  two  com- 
ponents, one  perpendicular  to  the  back  of  the  wedge,  and  the  other 
parallel  to  it.    The  latter  produces  no  effect.    The  same  is  true  of 
the  resistances ;  we  need  to  consider  only  those  components  of  them 
which  are  perpendicular  to  the  sides  of  the  wedge. 

Let  M  N  0  (Fig.  107)  represent  a  section  of 
the  wedge  perpendicular  to  its  faces ;  then  P  A, 
Q  A,  and  R  A,  drawn  perpendicular  to  the  faces 
severally,  show  the  directions  of  the  forces  which 
hold  the  wedge  in  equilibrium.  Taking  A  B  to 
represent  the  power,  draw  B  C  parallel  to  R  A, 
and  we  have  the  triangle  ABC,  whose  sides 
represent  these  forces.  But  A  B  C  is  similar  to 
M  N  0,as  their  sides  are  respectively  perpen- 
dicular to  each  other.  Hence,  calling  the  forces 
P,  Q,  and  R,  respectively, 

P:  Q::MN:MO; 


that  is,  there  is  equilibrium  in  a  wedge,  when 

The  power  is  to  the  resistances  as  the  lack  of 
the  wedge  to  the  sides  on  which  the  resistances  respectively  act. 

If  the  triangle  is  isosceles,  the  two  resistances  are  equal,  as  the 
proportions  show ;  and  P  is  to  either  resistance,  R,  as  the  breadth 
of  the  back  to  the  length  of  the  side. 


88  MECHANICS. 

If  tlie  resisting  surfaces  touch  the  sides  of  the  wedge  only  in 
one  point  each,  then  Q  A  and  R  A,  drawn  through  the  points  of 
contact,  must  meet  A  P  in  the  same  point  (Art.  60,  2) ;  otherwise 
the  wedge  will  roll,  till  one  face  rests  against  the  resisting  body 
in  two  or  more  points. 

The  efficiency  of  the  wedge  is  usually  very  much  increased  by 
combining  its  own  action  with  that  of  the  lever,  since  the  point 
where  it  acts  generally  lies  at  a  distance  from  the  point  where  the 
effect  is  to  be  produced.  Thus,  in  splitting  a  log  of  wood,  the  re- 
sistance to  be  overcome  is  the  cohesion  of  the  fibers ;  and  this  force 
is  exerted  at  a  distance  from  the  wedge,  while  the  fulcrum  is  a 
little  further  forward  in  the  solid  wood. 

VIII.  THE  K^TEE-JoiKT. 

140.  Description  and  Law  of  Equilibrium. — The  knee- 
joint  consists  of  two  bars,  usually  equal,  hinged  together  at  one 
end,  while  the  others  are  at  liberty  to  separate  in  a  straight  line. 
The  power  is  applied  at  the  hinge,  tending  to  thrust  the  bars  into 
a  straight  line ;  the  weight  is  the  force  which  opposes  the  separa- 
tion. 


Suppose  that  A  B  and  A  D  (Fig.  108)  are  equal  bars,  hinged 
together  at  A ;  and  that  the  bar  A  B  is  free  only  to  revolve  about 
the  axis  B,  while  the  end  D  of  the  other  bar  can  move  parallel  to 
the  base  E  F.  If  P  urges  A  toward  the  base,  it  tends  to  move  D 
further  from  the  fixed  point  B.  The  force  P'9  which  opposes  that 
motion,  is  represented  in  the  figure  by  the  weight  W.  The  law  of 
equilibrium  is, 

The  power  is  to  the  weight  as  twice  radius  to  the  tangent  of  half 
the  angle  between  the  bars. 

The  point  A  is  held  in  equilibrium  by  three  forces,  the  power 
P,  the  resistance  along  B  A,  and  that  along  D  A.  As  A  B  D  is 


THE    KNEE-JOINT.  89 

isosceles,  and  A  G  is  perpendicular  to  B  D,  the  angles  P  A  B  and 
PAD  are  equal ;  therefore  the  resistances  B  A  and  D  A  are 
equal  (Art.  61).  Let  a  =  the  angle  B  A  G  —\B  AD\  and  let 
R  =  the  resistance  in  the  line  D  A.  Then,  since  sin  P  A  £=sin  a, 

we  have 

P :  R  : :  sin  2  a  :  sin  a. 

But  W  is  equal  only  to  that  component  of  R  which  is  parallel  to 
B  D.  Therefore,  resolving  R,  we  have  (Art.  49), 

R  :  W : :  rad  :  cos  A  D  (7;  or 
: :  rad  :  sin  a. 

Compounding,  we  find, 

P  :  W : :  rad  .  sin  2  a  :  sin"  a. 

2  sin  a  .  cos  a     ... 

But  sin  2  a  = -3 ;  therefore 

rad 

_,    ^      2  rad  .  sin  a .  cos  a     . 
P:W''~  ~  :sinaa; 

rad 

,    rad  .  sin  a 

:  :  2  rad  : =  tan  a ; 

cos  a 

or  the  power  is  to  the  weight  as  twice  radius  to  the  tangent  of  the 
half  angle  between  the  bars. 

Since  A  G :  CD  : :  rad  :  tan  a, 

.'.P:W::2A  C:  CD;  or, 

The  power  is  to  the  weight  as  twice  the  height  of  the  joint  to 
half  the  distance  between  the  ends  of  the  bars. 

141.  Ratio  of  Power  and  Weight  Variable.— It  is  obvi- 
ous that  the  ratio  between  power  and  weight  is  different  for  differ- 
ent positions  of  the  bars.  As  A  is  raised  higher,  BAD  diminishes ; 
and  when  B  A  D  =  0,  then  a  =  0,  and  tan  a  =  0 ; 

.-.  P:  TF::2rad:  0, 

and  the  power  has  no  efficiency.  But  as  A  approaches  the  base 
E  F,  a  approaches  90° ;  therefore  tan  a  increases,  and  the  power  is 
more  -efficient.  When  A,  B,  and  D  are  in  a  straight  line,  a  —  90°, 
and  tan  a  is  infinite,  .*.  P  :  W : :  2  rad  :  oc .  Hence,  the  efficiency 
of  the  power  is  infinitely  great.  The  indefinite  increase  of  effi- 
.ciency  in  the  power,  which  occurs  during  a  single  movement,  ren- 
ders this  machine  one  of  the  most  useful  for  many  purposes,  as 
printing  and  coining. 

Questions  on  the  knee-joint. — 

1.  A  power  of  50  IDS.  is  exerted  on  the  joint  A  (Fig.  108) ; 
compare  the  weight  which  will  balance  it,  when  B  A  D  is  90°, 
and  when  it  is  160°.  Ans.  25  Ibs.  and  141.78  Ibs. 

2.  When  the  angle  between  the  bars  is  110°,  a  certain  power 


90  MECHANICS. 

just  overcomes  a  weight  of  65  Ibs. ;  what  must  be  the  angle,  in 
order  that  the  weight  overcome  may  be  five  times  as  great  ? 

Ana.  164°  3'  22". 

PRINCIPLE  OF  VIRTUAL  VELOCITIES. 

142.  The  Point  of  Application  Moving  in  the  Line  of 
the  Force. — In  examining  the  simple  machines,  we  have  in  each 
instance  simply  inquired  for  the  relative  magnitude  of  the  forces, 
called  the  power  and  the  weight,  when  in  equilibrium.  There  is 
another  important  particular  to  be  noticed,  namely,  the  relative 
velocity  of  the  power  and  weight,  when  they  begin  to  move.  It 
can  be  shown,  in  every  case,  that  the  velocities,  when  reckoned  in 
the  direction  in  which  the  forces  act)  are  inversely  as  the  forces. 

Some  examples  are  first  given  in  which  the  point  of  application 
moves  in  the  line  in  which  the  force  acts. 

In  the  straight  lever  (Fig.  109),  which  is  in  equilibrium  by  the 
weights  P  and  W,  suppose  a 

slight  motion  to  exist ;  then  FIG.  109. 

the  velocity  of  each  will  be 
as  the  arc  described  in  the 
same  time ;  but  the  arcs  are 
similar,  since  they  subtend 
equal  angles.  Therefore,  if  V  =  velocity  of  P,  and  v  =  velocity 

of  FT, 

V:v::AP:£W::A  CiB  0; 

but  it  has  been  shown  (Art.  106)  that 

P:W::B  C:A  (7; 
.-.  V:v  ::    W  :    P; 

that  is,  the  velocity  of  the  power  is  to  the  velocity  of  the  weight 
as  the  weight  to  the  power. '  Hence,  P  x  its  velocity  =  W  x  its 
velocity ;  that  is,  the  momentum  of  the  power  equals  the  mo- 
mentum of  the  weight. 

In  the  wheel  and  axle,  let  R  and  r  be  the  radii,  and  suppose  the 
machine  to  be  revolved ;  then  while  P  descends  a  distance  equal 
to  the  circumference  of  the  wheel  =  2  TT  R,  the  weight  ascends  a 
distance  equal  to  the  circumference  of  the  axle  =  2  n  r.  There- 
fore, 

V:v\  :2  n  R:  2  nr  :  :  R:r; 
but  (Art.  115),  P:W::  r  :  R; 

.-.  V:  v  ::  W\P\ 

or,  the  velocities  are  inversely  as  the  weights ;  and  P  x  V  =  W  x  v, 
the  momentum  of  the  power  equals  the  momentum  of  the  weight- 

In  the  fixed  pulley  the  velocities  are  obviously  equal ;  and  we 
have  before  seen  that  the  power  and  weight  are  equal ;  therefore 


PRINCIPLE    OF    VIRTUAL    VELOCITIES. 


01 


the  proportion  holds  true,  F:  v  : :  W:  P;  and  the  momenta  are 
equal. 

In  the  movable  pulley,  if  n  is  the  number  of  sustaining  parts 
of  the  cord,  when  W  rises  any  distance  =  x,  each  portion  of  cord  is 
shortened  by  the  distance  x,  and  all  these  n  portions  pass  over  to 
P,  which  therefore  descends  a  distance  =  n  x. 

Hence,  Viv::nx:x::n:l; 

but  (Art.  121),  P:  W::  1  in; 

.'.  V:v::W:P'} 
as  in  all  the  preceding  cases. 

In  the  screw  (Fig.  104),  while  the  power  describes  the  circum- 
ference —  2  TT  x  B  C,  the  weight  moves  only  the  distance  =  d ; 
therefore, 

F:  v:  :2  TT  x  B  C:  d; 
but  (Art.  133),  P  :  W: :  d  :  2  n  x  B  C; 

.:  V:v::  W:P; 

therefore  the  momentum  of  the  power  equals  the  momentum  of 
the  weight,  as  before. 

143.  The  Point  of  Application  Moving  in  a  Different 
Line  from  that  in  which  the  Force  Acts.— The  cases  thus 
far  noticed  are  the  most  obvious  ones,  because  the  points  of  appli- 
cation of  power  and  weight  actually  move  in  the  directions  in 
which  their  force  is  exerted.  But  the  principle  we  are  considering 
is  that  of  virtual  velocities.  If  the  force  is  exerted  in  one  line,  and 
the  motion  of  the  point  of  application  is  in  a  different  line,  then 
its  virtual  velocity  is  merely  that  component  which  lies  in  the 
former  line.  The  case  of  the  inclined  plane  will  illustrate  the 
principle. 

First,  let  P  (Fig.  110)  act  parallel  to  the  plane,  and  suppose 
the  body  to  be  moved  either  up  or  down  the  plane  a  distance  equal 
to  6r  d.  That  is  the  velocity 
of  the  power.  But  in  the  di- 
rection of  the  weight  (force  of - 
gravity),  the  body  moves  only 
the  distance  5  d.  Therefore 
the  velocity  of  the  power  is  to 
the  velocity  of  the  weight 
(each  being  reckoned  in  the 
line  of  its  action)  as  G  d  to 
bd. 

By  similar  triangles,    G  d:ld:\A  C:AB; 
or     F:    v  ::A  C:A  B. 

But  (Art.  127),  Pi   W  : :  A  B  :  A  C; 

.-.      F:    v  ::    W  :  P. 


FIG.  110. 


92  MECHANICS. 

Again,  let  the  power  act  in  any  oblique  direction,  as  G  e.  If 
the  body  moves  over  G  d,  draw  d  e  perpendicular  to  G  e ;  then  G  e 
is  the  distance  passed  over  in  the  direction  of  the  power,  and  ~b  d  in 
the  direction  of  the  weight.  G  d  being  taken  as  radius,  G  e  is 
cos  d  G  e  ^=  cos  (P  G  N  —  90°)  —  sin  P  G  N\  and  I  d  =  sin  a. 
Therefore,  the  virtual  velocity  of  the  power  is  to  the  virtual  velo- 
city of  the  weight  as  sin  P  G  N  to  sin  a ; 

or  F :  v  :    sin  P  G  N :  sin  a. 


But  (Art.  128),         P  :  W: 
.-.  F:  v  : 


sin  a  :  sin  P  G 
W:P. 


We  learn  from  the  foregoing  principle,  that  a  machine  does  not 
enable  us  to  obtain  any  greater  effect  than  the  power  could  pro- 
duce without  its  aid,  but  only  to  produce  an  effect  in  a  different 
form.  A  given  power,  for  instance,  may  move  a  much  greater 
quantity  of  matter  by  the  aid  of  a  machine,  but  it  will  move  it  as 
much  more  slowly.  On  the  other  hand,  a  power,  by  means  of  a 
machine,  may  produce  a  far  greater  velocity  than  would  be  possible 
without  such  aid ;  but  the  quantity  moved,  or  the  intensity  of  the 
force  exerted,  would  be  proportionally  less.  By  machines,  there- 
fore, we  do  not  increase  the  effects  of  a  power,  but  only  modify 
them. 

FEICTIOK  i^-  MACHINERY. 

144.  The  Power  and  "Weight  not  the  only  Forces  in 
a  Machine. — For  each  machine  a  certain  proportion  has  been 
given,  which  ensures  equilibrium.    And  it  is  implied  that  if  either 
the  power  or  the  weight  be  altered,  the  equilibrium  will  be  de- 
stroyed.   But  practically  this  is  not  true ;  the  power  'or  weight 
may  be  considerably  changed,  or  possibly  one  of  them  may  be  en- 
tirely removed,  and  the  machine  still  remain  at  rest.  The  obstruc- 
tion which  prevents  motion  in  such  cases,  and  which  always  exists 
in  a  greater  or  less  degree,  arises  from  friction ;  and  friction  is 
caused  by  roughness  in  the  surfaces  which  rub  against  each  other. 
The  minute  elevations  of  one  surface  fall  in  between  those  of  the 
other,  and  directly  interfere  with  the  motion  of  either,  while  they 
remain  in  contact.    Polishing  diminishes  the  friction,  but  can 
never  remove  it,  for  it  never  removes  all  roughness. 

As.  friction  always  tends  to  prevent  motion,  and  never  to  pro- 
duce it,  it  is  called  a  passive  force.  It  assists  the  power,  when  the 
weight  is  to  be  kept  at  rest,  but  opposes  it,  when  the  weight  is  to 
be  moved.  There  are  other  passive  forces  to  be  considered  in  the 
study  of  science,  but  no  other  has  so  much  influence  in  the  opera- 
tions of  machinery  as  friction. 

145.  Modes  of  Experimenting. — When  one  surface  slides 
on  another,  the  friction  which  exists  is  called  the  sliding  friction  ; 


FRICTION    IN    MACHINERY.  93 

but  when  a  wheel  rolls  along  a  surface,  the  friction  is  called  rolling 
friction.  The  sliding  friction  occurs  much  more  in  machines 
than  the  rolling  friction. 

Experiments  for  ascertaining  the  laws  of  friction  may  be  per- 
formed by  placing  on  a  table  a  block  of  three  different  dimensions, 
and  measuring  its  friction  un- 
der different  circumstances  by 
weights  acting  on  the  block  by 
means  of  a  cord  and  pulley,  as 
represented  in  Fig.  111.     This 
was  the  method  by  which  Cou- 
lomb first  ascertained  the  laws 
of  friction.  • 

Another  mode  is  to  place  the  block  on  an  inclined  plane,  whose 
angle  can  be  varied,  and  then  find  the  relative  friction  in  different 
cases,  by  the  largest  inclination  at  which  it  will  prevent  the  block 
from  sliding.     For,  when  W  on  the 
inclined  plane  A  B  (Fig.  112),  is  on 
the  point  of  sliding  down,  friction 
is  the  power,  which  acting  parallel 
to  the  plane,  is  in  equilibrium  with 
the    weight.     In    such    cases,    the 
power  is  to  the  weight  as  the  height 
to  the  length. 

The  coefficient  of  friction  is  the  fraction  whose  numerator  is 
the  force  required  to  overcome  the  friction,  and  its  denominator 
the  weight  of  the  body. 

146.  Laws  of  Sliding  Friction. — The  laws  of  sliding  fric- 
tion on  which  experimenters  are  generally  agreed  are  the  fol- 
lowing: 

1.  Friction  varies  as  the  pressure. — If  weights  are  put  upon 
the  block,  it  is  found  that  a  double  weight  requires  a  double  force 
to  move  it,  a  triple  weight  a  triple  force,  &c. 

2.  It  is  the  same,  however  great  or  small  the  surface  on  which  the 
lody  rests. — If  the  block  be  drawn,  first  on  its  broadest  side,  then  on 
the  others  in  succession,  the  force  required  to  overcome  friction  is 
found  in  each  case  to  be  the  same.    Extremes  of  size  are,  how- 
ever, to  be  excepted.    If  the  loaded  block  were  to  rest  on  three 
or  four  very  small   surfaces,  the  obstruction  might  be  greatly 
increased  by  the  indentations  thus  occasioned  in  the  surface 
beneath  them. 

3.  Friction  is  a  uniformly  retarding  force. — That  is,  it  destroys 
equal  amounts  of  motion  in  equal  times,  whatever  may  be  the 
velocity,  like  gravity  on  an  ascending  body. 


94  MECHANICS. 

4.  Friction  at  the  first  moment  of  contact  isjess  than  after  con- 
tact has  continued  for  a  time. — And  the  time  during  which  fric- 
tion increases,  varies  in  different  materials.    The  friction  of  wood  „ 
on  wood  reaches  its  maximum  in  three  or  four  minutes ;  of  metal 
on  metal,  in  a  second  or  two ;  of  metal  on  wood,  it  increases  for 
several  days. 

5.  Friction  is  less  "between  substances  of  different  kinds  than 
between  those  of  the  same  kind. — Hence,  in  watches,  steel  pivots 
are  made  to  revolve  in  sockets  of  brass  or  of  jewels,  rather  than 
of  steel. 

147.  Friction  of  Axes. — In  machinery,  the  most  common 
case  of  friction  is  that  of  an  axis  revolving  in  a  hollow  cylinder,  or 
the  reverse,  a  hollow  cylinder  revolving  on  an  axis.    These  are 
cases  of  sliding  friction,  in  which  the  power  that  overcomes  the 
friction,  usually  acts  at  the  circumference  of  a  wheel,  and  there- 
fore at  a  mechanical  advantage.    Thus,  the  friction  on  an  axis, 
whose  coefficient  is  as  high  as  20  per  cent.,  requires  a  power  of 
only  two  per  cent,  to  overcome  it,  provided  the  power  acts  at  the 
circumference  of  a  wheel  whose  diameter  is  ten  times  that  of 
the  axis. 

148.  Rolling  Friction. — This  form  of  friction  is  very  much 
less  than  the  sliding,  since  the  projecting  points  of  the  surfaces  do 
not  directly  encounter  each  other,  but  those  of  the  rolling  wheel 
are  lifted  up  from  among  those  of  the  other  surface,  as  the  wheel 
advances. 

By  the  use  of  the  apparatus  described  in  Art.  145,  the  laws  of 
the  rolling  are  found  to  be  the  same  as  those  of  the  sliding  fric- 
tion. But  on  account  of  the  manner  in  which  this  form  of  fric- 
tion is  overcome,  there  is  this  additional  law :  ^ 

The  force  required  to  roll  the  wheel  varies  inversely  as  the 
diameter. 

For  the  power,  acting  at  the  centre  of  the  wheel  to  turn  it  on 
its  lowest  point  as  a  momentary  fulcrum,  has  the  advantage  of 
greater  acting  distance  as  the  diameter  increases. 

It  is  the  rolling  friction  which  gives  value  to  friction  wheels, 
as  they  are  called.  When  it  is  desirable  that  a  wheel  should 
revolve  with  the  least  possible  friction,  each  end  of  its  axis  is  made 
to  rest  in  the  angle  between  two  other  wheels  placed  side  by  side, 
as  shown  in  Fig.  113.  The  wheel  is  obstructed  only  by  the  rolling 
friction  on  the  surfaces  of  the  four  wheels,  and  the  retarding  effect 
of  the  sliding  friction  at  the  pivots  of  the  latter  is  greatly  reduced 
on  the  principle  of  the  wheel  and  axle. 

The  sliding  friction  is  diminished  by  lubricating  the  surface, 
the  rolling  friction  is  not. 


MOTION    ON    INCLINED    PLANES.  95 

FIG.  113. 


149.  Advantages  of  Friction. — Friction  in  machinery  is 
generally  regarded  as  an  evil,  since*  more  power  is  on  this  account 
required  to  do  the  work  for  which  the  machine  is  made.  But  it  is 
easy  to  see,  that  in  general  friction  is  of  incalculable  value,  or 
rather,  that  nothing  could  be  accomplished  without  it.  Objects 
stand  firmly  in  their  places  by  friction  ;  and  the  heavier  they  are, 
the  more  firmly  they  stand,  because  friction  increases  with  the 
pressure.  All  fastening  by  nails,  bolts,  and  screws,  is  due  to  fric- 
tion. The  fibers  of  cotton,  wool  or  silk,  when  intertwined  with 
each  other,  form  strong  threads  or  cords,  only  because  of  the  power 
of  friction.  Without  friction,  it  would  be  impossible  to  walk  or 
even  to  stand,  or  to  hold  anything  by  grasping  it  with  the  hand. 


CHAPTER   VII. 

MOTION  ON  INCLINED  PLANES.— THE  PENDULUM. 

•150.  The  Force  which  Moves  a  Body  Down  an  In- 
clined Plane. — It  was  shown  (Art.  127)  that  when  the  power 
acts  in  a  line  parallel  to  the  inclined  plane,  P  :  W : :  A  B  :  A  C. 
If,  therefore,  P  ceases  to  act,  the  body  descends  the  plane  only 
with  a  force  equal  to  P. 

Let  g  (the  velocity  acquired  in  a  second  in  falling  freely)  =  the 
force  of  gravity,  /  —  the  force  acting  down  the  plane,  li  —  the 
height,  I  —  the  length ;  then  by  substitution, 


96  MECHANICS. 

fiffiih:  I,  and 


Therefore,  the  force  which  moves  a  body  down  an  inclined 
plane  is  equal  to  that  fraction  of  gravity  which  is  expressed  by  the 
height  divided  by  the  length.  This  is  evidently  a  constant  force 
on  any  given  plane,  and  produces  uniformly  accelerated  motion. 
Therefore  the  motion  on  an  inclined  plane*  does  not  differ  from 
that  of  free  fall  in  kind,  but  only  in  degree.  Hence  the  formulas 
for  time,  space,  and  velocity  on  an  inclined  plane  are  like  those 
relating  to  free  fall,  if  the  value  off  be  substituted  for  g. 

151.  Formulae  for  the  Inclined  Plane.  —  The  formulae  for 
free  fall  (Art.  28)  are  here  repeated,  and  against  them  the  corre- 
sponding formulae  for  descent  on  an  inclined  plane. 

Free  fall.  Descent  on  an  inclined  plane. 


2.      t  =  V— s  =  \/^. 

9  ffh 


V'Zgs v  = 


5.  *  =  v-. t  =  l" 

9  9^ 

6.  ,  =  gt ,=  £±1. 

li 

By  formula  1 ,  s  oc  P9  and  by  formula  3,  s  PC  va.  It  follows  that 
in  equal  successive  times  the  spaces  of  descent  are  as  the  odd 
numbers,  1,  3,  5,  &c.,  and  of  ascent  as  these  numbers  inverted ; 
also,  that  with  the  acquired  velocity  continued  uniformly,  a  body 
moves  twice  as  far  as  it  must  descend  to  acquire  that  velocity.  If 
a  body  be  projected  up  an  inclined  plane,  it  will  ascend  as  far  as  it 
must  descend  in  order  to  acquire  the  velocity  of  projection.  The 
distance  passed  over  in  the  time  t  by  a  body  projected  with  the 

velocity  v,  down  or  up  an  inclined  plane,  equals  t  v  ±  ^T . 

&  L 

These  statements  are  proved  as  in  the  case  of  free  fall,  Chapter  II. 

152.  Formulae  for  the  whole  Length  of  a  Plane. — 

1.  The  velocity  acquired  in  descending  a  plane  is  the  same  as 
that  acquired  in  falling  down  its  height.  ^ 

For  now  s  =  l\  hence  (formula  4),  v  =  (~j-— )   —  (2  g  £)f, 


DESCENT    ON    INCLINED    PLANES. 


which  is  the  formula  for  free  fall  through  h,  the  height  of  the 
plane. 

On  different  planes,  therefore,  v  oc  h^. 

2.  The  time  of  descending  a  plane  is  to  the  time  of  fatting  down 
its  height  as  the  length  to  the  height. 

For  (formula  2)  t  =  ^j)    =  I  (^j)        But  the  time  of 
fall  down  the  height  is  (  —  j  .    Therefore, 
t  down  plane  :  t  down  height  :  :  I  (—  y)  :  (  —  )  ; 

7/2\*     i/aV* 

::l  [-)  :  h(-\  : 

\gi      \g) 


On  different  planes,  t  oc  —  —  . 
V  h 

It  follows  that  if  several  planes  have  the  same  height,  the  veloc- 
ities acquired  in  descending  them  are  equal,  and  the  times  of 
descent  are  as  the  lengths  of  the  planes.  For,  let  A  C,  A  D,  A  E, 

(Fig.  114)  have  the  same  height  A  B',  then,  since  v  oc  ^,  and  7ns 
the  same  for  all,  v  is  the  same.  And  since  t  oc  ~—  ,  and  h  is  the 
same  for  all  the  planes,  t  oc  I. 


FIG.  114. 


D 


B 


153.  Descent  on  the  Chords  of  a  Circle. — In  descending 
the  chords  of  a  circle  which  terminate  at  the  ends  of  the  vertical 
diameter,  the  acquired  velocities  are  as  the  lengths,  and  the  times  of 
descent  are  equal  to  each  other  and  to  the  time  of  falling  through  the 
diameter. 

For  (Art.  152)  the  velocity  acquired  on  A  C  (Fig.  115)  = 


MECHANICS. 


=  A  0 


is  constant,  varies  as  A  C,  the  length. 

i 

(2  A  C*\^ 
-  A  —  )     ~ 
ff  •  A  c  I 

i  i 

\*      fiAB\*     ....  ....       - 

/   =  I  --  /  '  wnicn  1S  equal  to  the  time  of  falling 


g.Ac    I       \   g    / 
freely  through  A  B,  the  diameter. 

154.  Velocity  Acquired  on  a  Series  of  Planes. — If  no 

velocity  be  lost  in  passing  from  one  plane  to  another,  the  velocity 

acquired  in  descending  a  series  of  planes  is  equal  to  that  acquired 

in  falling  through  their  perpendicu-  1 

lar  height.     For,  in  Fig.  116,  the  A^  E        _F 

velocity  at  B  is  the  same,  whether 

the  body  comes  down  A  B  or  E  B, 

as  they  are  of  the  same  height,  F  b. 

If,  therefore,  the  body  enters  on  B  C 

with  the  acquired  velocity,  then  it  is 

immaterial  whether  the  descent  is 

on  A  B  and  B  C  or  on  E  (7;  in    D 

either  case,  the  velocity  at  C  is  equal  to  that  acquired  in  falling  Fc. 
In  like  manner,  if  the  body  can  change  from  B  G  to  CD  without 
loss  of  velocity,  then  the  velocity  at  D  is  the  same,  whether  ac- 
quired on  A  Bf  B  C,  and  G  D,  or  on  F  D,  which  is  the  same  as 
down  F  G. 

155.  The  Loss  in  Passing  from  one  Plane  to  Another. — 

The  condition  named  in  the  foregoing  article  is  not  fulfilled.  A 
body  does  lose  velocity  in  passing  from  one  plane  to  another.  And 
the  loss  is  to  the  whole  previous  velocity  as  the  versed  sine  of  the 
angle  between  the  planes  to  radius. 

Let  B  F  (Fig.  117)  represent  the  velocity  which  the  body  has 
at  B.  Eesolve  it  into  B  D  on  the  second  plane,  and  D  ^perpen- 
dicular to  it.  B  D  is  the  initial  velocity  on  B  (7;  „ 
and,  if  B  I  =  B  F,  D  I  is  the  loss.  But  D  I  is 
the  versed  sine  of  the  angle  F  B  D,  to  the  radius 
B  F\  and  /.  the  loss  is  to  the  velocity  at  B  as  D  I 
\BF\\  ver.  sin  B  :  rad. 

156.  No  Loss  on  a  Curve. — Suppose  now 
the  number  of  planes  in  a  system  to  be  infinite ; 
then  it  becomes  a  curve  (Fig.  118).    As  the  angle 
between  two  successive  elements  of  the  curve  is  in- 
finitely small,  its  chord  is  also  infinitely  small ;  but 


SIMILAR    SYSTEMS    OF    PLANES. 


99 


its  versed  sine  is  infinitely  smaller  still,  i.  e.,  an  infinitesimal  of  the 

second  order ;  for  diam. :  chord  : :  chord  :  ver.  FTQ 

sin.    Therefore,  although  the  sum  of  all  the 

infinitely  small  angles  is  a  finite  angle,  AOD, 

yet,  as  the  loss  of  velocity  at  each  point  is  an 

infinitesimal  of  the  second  order,  the  entire 

loss  (which  is  the  sum  of  the  losses  at  all 

the  points  of  the  curve)  is  an  infinitesimal  of 

i\\Q  first  order. 

Hence,  a  body  loses  no  velocity  on  a  curve,  and  therefore 
acquires  at  the  bottom  the  same  velocity  as  in  falling  freely 
through  its  height. 

It  appears,  therefore,  that  whether  a  body  descends  vertically, 
or  on  an  inclined  plane,  or  on  a  curve  of  any  kind,  the  acquired 
velocity  is  the  same,  if  the  height  is  the  same. 

157.  Times  of  Descending  Similar  Systems  of  Planes 
and  Similar  Curves. — If  planes  are  equally  inclined  to  the  hori- 
zon, the  times  of  describing  them  are  as  the  square  roots  of  their 
lengths.  For,  if  the  height  and  base  of  each  plane  be  drawn,  simi- 
lar triangles  are  formed,  and  h  :  I  is  a  constant  ratio  for  the  several 

planes.    By  Art.  152,  t  oc  — — -  oc  — -  oc  VT;  that  is,  the  time  va- 

vh        vl 

ries  as  the  square  root  of  the  length. 

If  two  systems  of  planes  are  similar,  i.  e.,  if  the  corresponding 
parts  are  proportional  and  equally  inclined  to  the  horizon,  it  is 
still  true  that  the  times  of  descending  them  are  as  the  square  roots 
of  their  lengths. 

Let  A  B  CD  and  a  I  c  d  (Fig.  119)  be  similar,  and  let  A  .Fand 


a  e    f 


a/be  drawn  horizontally,  and  the  lower  planes  produced  to  meet 
them,  then  it  is  readily  proved  that  all  the  homologous  lines  of 
the  figures  are  proportional,  and  their  square  roots  also  propor- 
tional. Then  (reading  t,  A  B,  time  down  A  B,  &c.), 

we  have  t,AB:t,al::  V~AB  : 

t,EB'.t,el: 


100  MECHANICS. 

and  t,EC\t,ec\ 

.'.  (by  subtraction)     t,  B  C :  t,b c: 

In  like  manner,  t,  CD  :  t,  cd  : 
.*.  (by  addition) 

t,(AB  +  BC  +  C  D):t,(a  b  +  bc  +  cd):i  VA£ :  Val 

: :  tf(A  B  +  B  C  +  CD)  :  \f(a  b  +  b  c  +  c  d. 

Though  there  is  a  loss  of  velocity  in  passing  from  one  plane  to 
another,  the  proposition  is  still  true ;  because,  the  angles  being 
equal,  the  losses  are  proportional  to  the  acquired  velocities ;  and 
therefore  the  initial  velocities  on  the  next  planes  are  still  in  the  same 
ratio  as  before  the  losses ;  hence  the  ratio  of  times  is  not  changed. 

The  reasoning  is  applicable  when  the  number  of  planes  in  each 
system  is  infinitely  increased,  so  that  they  become  curves,  similar, 
and  similarly  inclined  to  the  horizon.  Suppose  these  curves  to  be 
circular  arcs ;  then,  as  they  are  similar,  they  are  proportional  to 
their  radii.  Hence,  the  times  of  descending  similar  circular  arcs 
are  as  the  square  roots  of  the  radii  of  those  arcs. 

158.  Questions  on  the  Motions  of  Bodies  on  Inclined 
Planes. — 

1.  How  long  will  it  take  a  body  to  descend  100  feet  on  a  plane 
whose  length  is  150  feet,  and  whose  height  is  60  feet  ? 

Ans.  3.9  sec. 

2.  There  is  an  inclined  railroad  track,  2J  miles  long,  whose 
inclination  is  1  in  35.    What  velocity  will  a  car  acquire,  in  run- 
ning the  whole  length  of  the  road  by  its  own  weight  ? 

Ans.  106.2  miles  per  hour. 

3.  A  body  weighing  5  Ibs.  descends  vertically,  and  draws  a 
weight  of  6  Ibs.  up  a  plane  whose  inclination  is  45°.    How  far  will 
the  first  body  descend  in  10  seconds  ?  Ans.  3.44  feet. 

4.  A  body  descends  vertically  and  draws  another  body  of  half 
the  weight  up  an  inclined  plane.    When  the  bodies  had  described 
a  space  c  the  cord  broke,  and  the  smaller  body  continued  its  mo- 
tion through  an  additional  space  c  before  it  began  to  descend. 
What  is  the  inclination  of  the  plane  ?  Ans.  30°. 

159.  The  Pendulum. — A  pendulum  is  a  weight  attached  by 
an  inflexible  rod  to  a  horizontal  axis  of  suspension,  so  as  to  be  free 
to  vibrate  by  the  force  of  gravity.    If  it  is  drawn  aside  from  its 
position  of  rest,  it  descends,  and  by  the  momentum  acquired,  rises 
on  the  opposite  side  to  the  same  height,  when  gravity  again  causes 
its  descent  as  before.    If  unobstructed,  its  vibrations  would  never 
cease. 

A  single  vibration  is  the  motion  from  the  highest  point  on  one 
side  to  the  highest  point  on  the  other  side.  The  motion  from  the 


LENGTH    OF    A    P£tt J5MJ£ !?]£.'•  ^  i'<  ^  ;;  101 

J 


highest  point  on  one  side  to  the  same  point  again  is  called  a 
double  vibration. 

The  axis  of  the  pendulum  is  a  line  drawn  through  its  centre  of 
gravity  perpendicular  to  the  horizontal  axis  about  which  the  pen- 
dulum vibrates. 

The  centre  of  oscillation  of  a  pendulum  is  that  point  of  its  axis 
at  which,  if  the  entire  •  mass  were  collected,  its  time  of  vibration 
would  be  unchanged. 

The  length  of  a  pendulum  is  that  part  of  its  axis  which  is 
included  between  the  axis  of  suspension  and  the  centre  of  oscil- 
lation. 

All  the  particles  of  a  pendulum  may  be  conceived  to  be  col- 
lected in  points  lying  in  the  axis.  Those  which  are  above  the  cen- 
tre of  oscillation  tend  to  vibrate  quicker  (Art.  157),  and  therefore 
accelerate  it;  those  which  are  below  tend  to  vibrate  slower,  and 
therefore  retard  it.  But,  according  to  the  definition  of  the  centre 
of  oscillation,  these  accelerations  and  retardations  exactly  balance 
each  other  at  that  point. 

160.  Calculation  of  the  Length  of  a  Pendulum. — Let 

C  q  (Fig.  120)  be  the  axis  of  a  pendulum  in  which  all  its  weight 
is  collected,  G  the  point  of  suspension,  G  the  centre  of 
gravity,  0  the  centre  of  oscillation,  a,  b,  &c.,  particles   FlG- 12°- 
above  0,  which  accelerate  it,  p,  q,  &c.,  particles  below  0,     ~T"  c 
which  retard  it.     C  0  ^=  I,  is  the  length  of  the  pendulum 
required.    Denote  the  masses  concentrated  in  a,  b  . .  . 
p,  q,  by  m,  m' .  . . .  m"  mfrf,  and  their  distances  from  C  by 
r,  rr .  . . .  r",  r'"\  and  denote  the  distance  from  G  to  G 
by  Tc.    Denote  the  angular  velocity  by  0 ;  then  the  ve- 
locity of  m  will  be  r  6  and  its  momentum  will  be  m  r  0. 

If  m  had  been  placed  at  0,  the  moving  force  would 
have  been  m  16.  The  difference  m  (I  —  r)  6,  is  that  por- 
tion of  the  force  which  accelerates  the  motion  of  the 
system. 

The  moment  of   this  force  with    respect  to   C  is     "  " 
m  (I  -  r)  r  0. 

In  like  manner  the  moment  of  m'  is  m'  (I  —  r1)  r'  0, 
and  so  on  for  all  the  particles  between  C  and  0. 

The  moments  of  the  forces  tending  to  retard  the  sys- 
tem applied  at  the  points  p,  q,  &c.,  are 

m"  (r"  -  1)  r"  0,  m'"  (rm  -  I)  r'"  0,  &c. 
But  since  these  forces  are  to  balance  each  other,  we  have 

m  (l  -  r)  r  6  +  m'  (I  -  r1)  r'  d  +  &c.  =  m"  (r"  -  I)  r"0 


whence  2  =  m'  r"  +  m"  r"*  +  &C> 

m  r  4-  m'  r1  +  w"  r"  +  &c. 

8  (m  r9) 

Or  I  =  -^  -  ~,  where  $  denotes  the  sum  of  all  the  terms  similar 
S  (m  r)  ' 

to  that  which  follows  it. 

The  numerator  of  this  expression  is  called  the  moment  of  inertia 
of  the  hody  with  respect  to  the  axis  of  suspension,  and  the  denomi- 
nator is  called  the  moment  of  the  mass,  with  respect  to  the  axis  of 
suspension. 

By  the  principle  of  moments  (Art.  77)  m  r  +  m'  r'  +  &c.,  or 
S  (m  r)  =  M  k,  where  M  denotes  the  entire  mass  of  the  pendulum  ; 

,      8  (m  r2) 
hence,  by  substitution,  I  =  —  -M  -,    • 

JM.  /C 

That  is,  the  distance  from  the  axis  of  suspension  to  the  centre 
of  oscillation  is  found  by  dividing  the  moment  of  inertia,  with 
respect  to  that  axis,  by  the  moment  of  the  mass  with  respect  to  the 
same  axis. 

161.  The  Point  of  Suspension  and  the  Centre  of  Os- 
cillation Interchangeable.  —  Let  the  pendulum  now  be  sus- 
pended from  an  axis  passing  through  0,  and  denote  by  I'  the 
distance  from  0  to  the  new  centre  of  oscillation.  The  distances  of 
a,b....p,q,  from  0,  will  be  I  —  r,  I  —  r',  &c.,  and  the  distance 
Q  Owillbe7-&. 

Hence,  from  the  principle  just  established,  we  have 

7,  _  8[m(l  —  r)2]  __  8(mF  -Zmrl  +  mr3) 
M(l-k)  M(l-lc) 

S(m?}  -2S(mrl)  +  S(mr*) 
M(l-  k) 

But  8  (m  r2)  =  M  Tc  I  ;  and  since  I  is  constant, 

_  M?  -2l8(mr)  +  MJcl  _  M?  -  2  M  k  I  +  M  k  I 

M(l-k)  M(l-k) 

_  M(l-Jc)  I  _  ' 

'  ' 


This  last  equation  shows  that  the  centre  of  oscillation  and  the 
point  of  suspension  are  interchangeable  ;  that  is,  if  the  pendulum 
were  suspended  from  0,  it  would  vibrate  in  the  same  time  as  when 
suspended  from  (7. 

162.  The  Cycloid.  —  One  of  the  methods  of  investigating  the 
theory  of  the  pendulum  is  by  means  of  the  properties  of  the  cy- 
cloid. This  curve  is  described  by  a  point  situated  on  the  circum- 
ference of  a  circle,  as  it  rolls  on  a  straight  line. 

Let  the  circled  H  B  (Mg.  121)  make  one  revolution  upon  the 


PROPERTIES  OP  THE  CYCLOID. 


103 


line  G  A  X,  equal  to  its  circumference  ;  the  curve  line  C  D  B  X9 

traced  out  by  that  point  of  the  circle  which  was  in  contact  with 

C  when  the  circle  began  to 

roll,  is  called  a  cycloid.    If 

C  X  be  bisected  in  A,  and 

A  B  be  drawn  at  right  angles 

to  it,  it  is  evident,  from  the 

manner  in  which  the  curve 

is  generated,  that  it  will  have 

similar    branches    on    both 

sides  of  A  B,  and  that  its  vertex  B  will  be  so  placed  as  to  make 

the  axis  A  B  equal  to  the  diameter  of  the  generating  circle.    The 

properties  of  the  cycloid,  as  applied  to  the  vibration  of  the  pendu- 

lum, are  the  following. 

163.  The  Cycloidal  Ordinate  D  H  equals  the  circular  arc 
B  H.  —  For,  let  ft  Da  (Fig.  121)  be  the  position  of  the  circle  when  the 
generating  point  is  at  D  ;  draw  the  diameter  b  a  parallel  to  B  A, 
and  from  D  draw  DHL  parallel  to  C  A  ;  then  the  arc  Da—  arc 
HA,  :.  the  sines  D  0,  H  L,  are  equal;  hence  D  H  =  0  L\  but 
from  the  mode  in  which  the  cycloid  is  generated,  C  a  —  arc  D  a, 
and  C  A  =  semi-circumference  B  HA  ;  hence  D  H  —  0  L  =  a  A 
—  C  A  —  C  a  —  semi-circumference  B  HA  —  arc  HA  =  arc  B  H. 


FIG.  122. 


164.  A  Tangent  to  the  Cycloid  at  any  point,  E  (Fig. 

is  parallel  to  the  corresponding  chord  B  K  of  the  generating  circle.  — 

Draw  DHL  infinitely  near  to  E  K  M  ; 

join  B  K,  and  produce  it  to  k.    The 

elementary  triangle  H  K  k  is  similar  to 

the  triangle  K  R  B  formed  by  the  tan- 

gents (K  R,  B  R)  to  the  circle  at  the 

points  .ZT,  B,  and  is  consequently  isos- 

celes ;  /.  KH  =  H  Ik.    Now  (Art.  163), 

arc  B  KH  =  D  H  ;  from  which  equation 

subtract  the  previous  one,  and  arc  B  K 

-Die.    But  arc  B  K  =  E  K;  :.  E  K   v 

=  D  k.    Hence,  since  E  K  and  D  Ic  are  equal  and  parallel,  E  D  and 

K  k  must  also  be  equal  and  parallel  ;  and  as  the  tangent  at  the 

point  E  may  be  considered  as  coinciding  with  E  D,  it  must 

therefore  be  parallel  to  the  chord  B  K. 

Hence  the  ends  of  the  cycloid  meet  the  base  at  right  angles; 
for  the  tangent  at  C  is  parallel  to  B  A,  the  axis. 

165.  The  Cycloidal  Arc  B  E  is  equal  to  twice  the  correspond- 
ing chord  B  K  of  the  generating  circle.  —  Draw  H  o  perpendicular 
to  K  k  ;  and  since  the  triangle  K  H  k  is  isosceles,  H  o  bisects  the 


101- 


MECHANICS. 


base  KJc,.\KJcGYED  =  2  K  o ;  and  since  H  o  may  be  consid- 
ered as  a  small  circular  arc  described  with  radius  B  H,  K  o  —  Bo 
—  B  K  —  B  H  —  B  K\  hence  E  D  and  K  o  are  corresponding 
increments  of  the  cycloidal  arc  B  E  and  the  chord  B  K\  and  as 
the  arc  and  chord  begin  together  from  the  point  B,  and  every  in- 
crement of  the  former  is  twice  the  corresponding  increment  of  the 
latter,  the  arc  B  E  must  be  equal  to  twice  the  chord  B  K\  conse- 
quently, the  whole  arc  B  C  =  twice  the  diameter  A  B ;  and  the 
length  of  the  whole  curve  C  B  X  (Fig.  121)  =  4  A  B.  And  as 
CA  X=  TT  .  A  B,  therefore  the  whole  cycloid  :  its  base  : :  4  :  TT. 

166.   Descent  by  Gravity  on  a  Cycloid— E  F  M  (Fig. 
123)  is  a  circle  whose  di- 
ameter E  Mis  perpendic-  FlG-  123- 
ular  to  the  horizon,  and 
B  G  M    is    the     corre- 
sponding semicycloid. 

Let  the  body  begin  to 
descend  from  any  point  .4. 

Draw  A  D  parallel  to 
B  E,  and  upon  M  D  as  a 
diameter  describe  the  cir- 
cle D  N  P  My  with  its 
centre  at  0. 

Put  h  =  D  M,  r  =  C  E,  x  =  D  H\  then  the  time  of  describing 


K. 


the  elementary  arc  G  K  will  be 


G  K        GK 


A 

and 


^     .    .,     _      , 
Bysimilar  triangles 

ON 


GK      FM     VMH7ME 


PQ     NH 

dividing  (1)  by  (2), 

G  K     VME.DH 


ON         ,       -       ^T      b/vi. 

--  .  (2)    Now  EL  —  P  0:  hence, 

VMH.DH 


ON 


NP~ 
GK=l 


and  the  time  of  descending  G  K  is 

o 

tVZrx.  NP 


whence 


=  NP 

' 


In  like  manner,  the  time  of  describing  any  other  elementary 

2     /r 

arc  will  be  found  to  be  T  y  -  times  the  corresponding  arc  on  the 
/&      g 


DESCENT    ON    A    CYCLOID. 


105 


circumference  D  NP  M;  hence  the  time  of  describing  the  cycloid- 

al    arc  A  M  will    be   f  |/-  x  arc  D  NP  M  =  ~  I/-  .  ^  = 
h  r  g  li  r  g     2 


FIG.  124 
E 


S 


This  expression  for  the  time  down  A  M  being  independent  of 
h,  is  very  remarkable,  for  it  proves  that 

The  time  of  descent  on  a  cycloid  to  the  lowest  point  is  alivays  the 
same,  from  whatever  point  in  the  curve  the  body  begins  to  descend. 

The  .time  of  falling  through  E  M  is  2  y  -;  .-.  time  down  A  M 

y 

:  time  down  E  Mint  y  -  :  2  V  -  ::  ir  :2  :  :  semi-circumference  : 

9        *  9 
diameter. 

167.  The  Involute  of  a  Semicycloid.  —  The  involute  of 
any  curve  is  another  curve  described  by  the  extremity  of  a  tangent 
as  it  unwinds  from  the  former,  which 
is  called  the  evolute.  If,  for  example, 
a  tangent  of  a  circle  unwinds  from  it, 
the  circumference  of  the  circle  is  the 
evolute,  and  the  spiral  described  by  the 
end  of  the  tangent  is  the  involute  of 
the  circle.  The  involutes  of  most 
curves  are  different  from  their  evo- 
lutes;  but  in  the  case  of  the  seini- 
cycloid,  the  involute  and  evolute  are 
of  the  same  form  and  size. 

Take  any  line  8  C  (Fig.  124),  and 
draw  8  A  at  right  angles  to  it  ;  make 
S  C  :  S  A  :  :  semi-circumference  of  a 
circle  :  its  diameter;  and  complete  the 
parallelogram  SODA.  Produce  S  A  to  B,  making  A  B  =  8  A  ; 
upon  S  C,  A  D,  describe  two  semicycloids  8  D,  D  B,  the  vertex  of 
the  former  of  which  is  at  D,  and  the  latter  at  B  ;  then  if  the  tan- 
gent unwinds,  beginning  at  D,  until  the  point  of  contact  reaches 
S,  its  extremity  will  always  be  found  in  the  semicycloid  D  B.  For, 
through  any  point  F  on  A  D,  draw  E  F  G  perpendicular  to  8  C, 
and  through  B  draw  B  G  parallel  to  S  <7;  then  EG—  SB;  on 
E  F,  F  G,  describe  the  semicircles  E  T  F,  F  P  G,  and  draw  the 
chords  T  F,  F  P,  the  former  of  which  (Art.  164)  is  a  tangent  to 
the  cycloid  8  D  at  T.  Now  8  E  =  arc  E  T,  and  S  G  =  E  T  F; 
:.  OE  (=  D  F)  =  arc  T  F;  but  D  F  =  F  P;  :.  arcs  F  T,  F  P 
are  equal,  and  also  the  angles  subtended,,  T  E  F,  F  G  P.  There- 


100 


MECHANICS. 


fore,  as  Tand  P  are  right  angles,  EF  T  =  P  F  G,  and  P  F  Tis 
a  straight  line;  moreover,  T  P  —  2  T  F  =  (Art.  165)  the  cycloidal 
arc  T  D.  Therefore,  T  P  is  a  tangent  unwound  from  D,  and  P 
is  its  extremity ;  and  P  having  been  assumed  anywhere  on  the 
semicycloid  D  B,  it  follows  that  D  P  B  is  the  involute  of  8  T  D. 

168.  The  Cycloidal   Pendulum.— A  pendulum  may  be 
made  to  vibrate  in  a  cycloid  by  attaching  the  weight  P  (Fig.  125) 
to  a  flexible  cord,  whose 

point  of  suspension  is  at 
$,  where  two  semicy- 
cloids  meet.  The  cord 
and  the  semi-cycloid 
should  be  of  the  same 
length,  and  then  (Art. 
167)  the  weight  P  will, 
at  each  vibration,  de- 
scribe arcs  of  the  cy- 
cloid D  B  E,  as  involutes 
of  SD  and  SR  Hence, 
the  conclusion  arrived  at 
in  Art.  166  applies  to  tins 

motion ;  namely,  the  time  down  P  B  from  any  point  P  :  time  down 
A  B  : :  semi-circumference  :  diameter;  .'.  doubling  the  antecedents, 
the  time  of  a  single  vibration  :  time  of  falling  half  the  length  of  the 
pendulum  : :  TT  :  1. 

169.  Application  to  the  Circular  Pendulum.— Since  the 

proportion  at  the  close  of  the  foregoing  article  is  always  true,  from 
whatever  point  the  descent  commences,  therefore 

All  the  vibrations  of  a  cycloidal  pendulum  are  performed  in 
equal  times,  however  large  or  small  the  extent  of  siving. 

This  is  not  true  of  any  other  curve.  But  it  is  evident  that  a 
very  short  arc  of  a  cycloid  at  the  lowest  point  B  is  coincident  with 
the  arc  of  a  circle  whose  centre  is  S.  Hence,  if  a  pendulum  vibrate 
through  very  short  arcs,  the  conclusions  are  practically  true,  that 
in  circular  pendulums  also  unequal  arcs  are  described  in  equal 
times,  and  that  the  time  of  a  vibration  is  to  the  time  of  falling 
through  half  the  length  of  the  pendulum  as.  TT  is  to  1.  For  this 
reason,  the  pendulum  of  an  astronomical  clock  is  so  connected 
with  the  machinery  by  its  scapement,  as  to  vibrate  in  small  arcs. 

170.  Relation  of  Time,  Length,  and  Force  of  Gravity.— 

Let  I  —  the  length  of  a  pendulum,  that  is,  the  distance  from  the 
point  of  suspension  to  the  centre  of  oscillation.  Then  the  time  of 


QUESTIONS    ON    THE    PENDULUM.  107 

i  i 

(2$\~      /A~ 
— )  =  (-)  .    Hence,  putting  t  =  time 

of  a  single  vibration, 


Therefore,  the  length  of  a  pendulum  being  known,  the  time  of 
one  vibration  is  found ;  and  on  the  other  hand,  if  the  time  of  a 
vibration  is  known,  the  length  of  the  pendulum  is  obtained  from 
it. 

From  the  same  formulae,  we  find  that  t  oc  Vl,  or 

The  time  in  which  a  pendulum  makes  a  vibration  varies  as  the 
square  root  of  the  length. 

As  t  x  Vl,  :.  I  x  t* ;  hence,  if  the  length  of  a  seconds  pendulum 
equals  I,  then  a  pendulum  which  vibrates  once  in  two  seconds 
equals  4  I,  and  one  which  beats  half  seconds  —  -}  I,  &c. 

Again,  by  observing  the  length  of  a  pendulum  which  vibrates 
in  a  given  time,  the  force  of  gravity,  g,  may  be  found.  For,  as  I  = 

^4>  •'•  g  —  -ia-»    And  if  g  varies,  as  it  does  in  different  latitudes 

If 

n  /^  * 

and  at  different  altitudes,  then  I  =  ^-5-  x  g  f ;  and  if  the  time  is 

constant  (as,  for  example,  one  second},  then  I  x  g.    Hence, 

The  length  of  a  pendulum  for  beating  seconds  varies  as  the  force 
of  gravity. 

Also,  t  x  ( — )  ;  that  is,  the  time  of  a  vibration  varies  directly 

as  the  square  root  of  the  length,  and  inversely  as  the  square  root 
of  the  force  of  gravity. 

Since  the  number,  n,  of  vibrations  in  a  given  time  varies  in- 

i 


versely  as  the  time  of  one  vibration,  therefore  n  oc      -    ,  and 

g  x  I  n\    Hence,  if  the  time  and  the  length  of  a  pendulum  are 
given, 

The  force  of  gravity  varies  as  the  square  of  the  number  of  vibra- 
tions. 

1.  What  is  the  length  of  a  pendulum  to  beat  seconds,  at  the 
place  where  a  body  falls  16^  ft.  in  the  first  second  ? 

Ans.  39.11  inches,  nearly. 

2.  If  39.11  inches  is  taken  as  the  length  of  the  seconds  pendu- 
lum, how  long  must  a  pendulum  be  to  beat  10  times  in  a  minute  ? 

Ans.  117J  feet. 

3.  In  London,  the  length  of  a  seconds  pendulum  is  39.1386 


108 


MECHANICS. 


inches ;  what  Telocity  is  acquired  by  a  body  falling  one  second  in 
that  place  ?  Ans.  32.19  feet. 

171.  The  Compensation  Pendulum. — This  name  is  given 
to  a  pendulum  which  is  so  constructed  that  its  length  does  not 
vary  by  changes  of  temperature.  As  all  substances  expand  by 
heat,  and  contract  by  cold,  therefore  a  pendulum  will  vibrate  more 
slowly  in  warm  than  in  cold  weather.  This  difficulty  is  overcome 
in  several  ways,  but  always  by  employing  two  substances  whose 
rates  of  expansion  and  contraction  are  un- 
equal. One  of  the  most  common  is  the  grid- 
iron pendulum,  represented  in  Fig.  126.  It 
consists  of  alternate  rods  of  steel  and  brass, 
connected  by  cross-pieces  at  top  and  bottom. 
The  rate  of  longitudinal  expansion  and  con- 
traction of  brass  to  that  of  steel  is  about  as 
100  to  61 ;  so  that  two  lengths  of  brass  will 
increase  and  diminish  more  than  three  equal 
lengths  of  steel.  Therefore,  while  there  are 
three  expansions  of  steel  downward,  two  up- 
ward expansions  of  brass  can  be  made  to  neu- 
tralize them.  In  the  figure  the  dark  rods  rep- 
resent steel,  the  white  ones  brass.  Suppose 
the  temperature  to  rise,  the  two  outer  steel 
rods  (acting  as  one)  let  down  the  cross-bar  d\ 
the  two  brass  rods  standing  on  d  raise  the  bar 
"b ;  the  steel  rods  suspended  from  Z>  let  down 
the  bar  0,  on  which  the  inner  brass  rods  stand, 
and  raise  the  short  bar  c;  and  finally,  the 
centre  steel  rod,  passing  freely  through  d  and  e,  lets  down  the  disk 
of  the  pendulum.  These  lengths  (counting  each  pair  as  a  single 
rod)  are  adjusted  so  as  to  be  in  the  ratio  of  100  for  the  steel  to  61 
for  the  brass ;  in  which  case  the  upward  expansions  just  equal 
those  which  are  downward,  and  therefore  the  centre  of  oscillation 
remains  at  the  same  distance  from  the  point  of  suspension. 

If  the  temperature  falls,  the  two  contractions  of  brass  are  equal 
to  the  three  of  steel,  so  that  the  pendulum  is  not  shortened  by 
cold. 

The  mercurial  pendulum  consists  of  a  steel  rod  terminating  at 
the  bottom  with  a  rectangular  frame  in  which  is  a  tall  narrow  jar 
containing  mercury,  which  is  the  weight  of  the  pendulum.  It 
requires  only  6.31  inches  of  mercury  to  neutralize  the  expansions 
and  contractions  of  42  inches  of  steeL 


PROJECTILES. 


109 


FIG.  127. 


CHAPTER   VIII. 

PROJECTILES   CENTRAL  FORCES. 

172.  Path  of  a  Projectile.— It  has   been  shown  already 
(Art.  44),  that  a  body  projected  in  any  direction  not  coincident 
with  the  vertical,  describes  a  parabola.    In  swift  motions,  however, 
the  path  of  a  projectile  differs  widely  from  a  parabola ;  and  the 
laws  of  atmospheric  resistance  must  be  employed  to  obtain  cor- 
rections for  the  conclusions  deduced  in  this  chapter. 

173.  Formulae  Investigated.— In  order  to  investigate  the 
general  formula,  let  A  (Fig.  127)  be  the  point  of  projection,  A  B 
the  plane  over  which  the  body  is  projected,  passing  through  A. 
A  B  also  denotes  the  range  or  dis- 
tance to  which  the  body  is  thrown. 

Let  A  C  be  drawn  parallel,  and 
BCD  perpendicular  to  the  hori- 
zon. Put  a  =  C  A  D,  the  angle  of 
elevation  ;  b  =  C  A  B,  the  angle  of 
elevation  or  depression  of  the  plane 
of  the  range ;  v  =  the  velocity  of 
projection  ;  t  =  the  time  of  flight ; 
r  —  the  range ;  and  g  =  32J  feet, 
the  velocity  imparted  by  gravity  in 
one  second. 

Then,  by  the  laws  of  uniform  motion,  at  the  end  of  the  time  #, 
if  gravity  did  not  act,  the  body  would  be  found  in  the  point  I), 
while,  by  the  laws  of  falling  bodies,  it  would  in  the  same  time  pass 
through  the  perpendicular  D  B ;  consequently, 

A  D  =  tv ;  and  D  B  =  J  g  t\ 

In  the  right-angled  triangles  ABC  and  ADC,  the  angle  B 
is  the  complement  of  b,  and  the  angle  D  is  the  complement  of  a ; 
and,  since  the  sides  are  as  the  sines  of  the  opposite  angles, 


cos 


I :  sin  (a  ±  b):-.tv:tv-n(a^b)  =  \gt\ 


cosb 


Phis  is  used  when  the  plane  A  B  descends ;  minus,  when  it 
ascends. 

g  t  %=  sin  (a  ±  b)  . 


Or, 


Again,    cos  a  :  sin  (a  ±  b) : :  r  : 


cos  b 
r  sin  (a  ±  b) 


cos  a 


110  MECHANICS. 

Or  gt*  _sm(a±t)  ,. 

27~     '    COB  a 
Eliminating  t  from  (1)  and  (2),  we  have 

r    _  2  sin  (a  ±  b)  cos  a 

~tf  ~  g  cos1  b          ••••••    (a) 

From  these  three  equations,  all  the  relations  "between  the  time, 
velocity,  range,  and  angle  of  elevation,  are  readily  determined; 
so  that  any  two  of  these  four  quantities  being  given,  the  other  two 
may  be  found.  Thus, 

By  equation  (1)         *  =  rffg^- 

T>  J.'         /n\  t*  COS  a 

By  equation  (2)         r  = 


.  . 

2  sin  (a  =fc  b) 

The  range  and  elevation  being  given,  to  find  the  time  and 
velocity. 

_,  /2  r  sin  (a  ±  b)\2 

By  equation  (2)          £  =  I  —    —  —  x  —  -  J  . 
V      g  cos  a)       / 

/o\  I        r  9  cos2  *        \* 

By  equation  (3)         v  =  I  -  —  :  —  ,     .   ..  -  1  . 

\2  sin  (a  db  £)  cos  07 

The  velocity  and  elevation  being  given,  to  find  the  &'we  and 


..••*-*, 

By  equation  (1)          ^  =  -   ---^  -  '-. 

g  cos  b 

,.      /ox  2  va  sin  (a  ±  b)  cos  « 

By  equation  (3)          r  —  -  L  —  ^~  -  • 

g  cos2  b 

If  any  two  of  the  above  quantities  be  given  to  find  the  angle  of 
elevation,  then  (b  being  known)  in  order  to  find  the  value  of  a  we 
substitute  in  formulae  (1),  (2),  and  (3),  for  sin  (a  d=  #),  its  value, 

O|T1 

viz.,  sin  a  cos  b  ±  sin  b  cos  a,  and,  in  reducing,  put  tan  for  —  . 

cos 

Formula  (1)  becomes  sin  a  ±  tan  b  cos  a  —  —  -,  whence  by 

eliminating  cos  a,  the  value  of  sin  a  can  be  found.  The  resulting 
equation  being  a  quadratic,  there  will  be,  in  general,  two  values 
of  sin  a  ;  that  is,  two  angles  of  elevation  for  the  same  value  of 
v  and  t. 

Formula  (2)  becomes  tan  a  cos  b  ±  sin  b  =  —-,  whence 

tan  a  —  ^—  -  -  r  qp  *an  &• 
2  r  cos  5  n 

Formula  (3)  becomes  sin  #  cos  a  ±  tan  5  cosa  a  =  ^  r   °f  -. 


ELEVATION    AND    RANGE.  HI 


Put  c  =  *  —  9  an(*  x  ~  s^n  a>  ^en  (1  ~~  s^)    =  cos  0;  and 


#  (1  —  #2)2  ±  tan  5  (1  —  a;2)  =  c,  from  which  a?  or  sin  «  may  be 
found. 

174.  Different  Angles  of  Elevation  for  the  Same 
Range.  —  As  this  last  equation  is  a  biquadratic,  it  will  give  four 
yalues  of  x  ;  the  two  positive  values  indicate  that  there  are  two 
different  angles  of  elevation  corresponding  to  the  same  values  of 
v  and  r.  When  these  two  values  are  equal,  then,  as  shown 
below,  a  =  £  (90°  qp  b),  in  which  cas*e  the  range  (r)  is  a  maxi- 
mum; and  there  is  the  same  range  for  any  two  angles  equally 
above  and  below  that  which  gives  the  maximum.  For,  since 

2  sin  (a  ±  b)  cos  a  .f         ,  ,  , 
r  —  2  v2  -  —^7  --  ,  if  v  and  the  angle  b  are  given,  the  range 

g  COS    0 

will  vary  as  sin  (a  ±  b)  cos  a.    But 

sin  (a  ±  b)  cos  a  =  sin  a  cos  a  cos  5  ±  sin  b  cos2  # 

•         —  J  sin  2  a  cos  5  ±  sin  b  (A  +  £  cos  2  a) 

=  ±  sin  2  w  cos  #  ±  £  cos  2  a  sin  5  ±  1  sin  b 
=  ism(20±£)dbjsin&; 

and  since  the  second  part  of  this  expression  is  constant,  the  range 
will  be  a  maximum  when  sin  (2  a  ±  b)  is  a  maximum  ;  that  is, 
when  2  a  ±  b  =  90°. 

/.  a  =  J  (90°  rp  5). 

Therefore  the  range  will  be  a  maximum  when  the  angle  of  eleva- 
tion is  equal  to  £  (90°  =F  b). 

TTTI.  1  /AAO       z\  2  sin  (90°  +  2  c)  ±  sin  b 

When  a  =  |  (90°  T  5)  +  c,        r  =  v*  -  *  —    —TT~      —, 

g  cos2  b 

,  sin  (90°  —  2  c)  =fc  sin  £ 

and  when  a  =  4-  (90°  ^  5)  —  c,        r  =  v*  -  -  -  r4  --  . 

g  cos2  b 

But  sin  (90°  +  2  c)  =  sin  (90°  —  2  c),  since  the  sines  of  sup- 
plementary arcs  are  equal  ;  hence  all  angles  of  elevation,  equally 
above  and  below  that  which  gives  the  maximum,  have  equal  ranges. 
Thus,  a  cannon  ball  fired  at  an  angle  of  60°  above  a  horizontal 
plane,  would  reach  the  plane  at  the  same  distance  from  the  point 
of  projection  as  if  fired  at  an  angle  of  30°.  When  the  data  of  the 
problem  give  or  require  a  greater  value  for  sin  (2  a  ±  b)  than  1, 
the  sine  of  90°,  the  problem,  under  the  proposed  conditions,  is 
impossible. 

That  the  two  values  of  sin  a  are  equal  when  the  range  is  a 
maximum,  may  be  shown  as  follows  : 

Let  x  and  y  be  two  varying  supplementary  arcs,  and  let 

2  sin  x  ±  sin  b 
> 


112  MECHANICS. 

sm  V  ^  sin 


i\     j/u  2 

Again,  let  a  =  ^  (u  re  b)  ;  then  r  =  v* 


—  _ 

g  cos2 


Now,  although  these  two  values  of  a  may  be  different,  yet  the 
ranges  corresponding  to  them  will  be  equal,  because  sin  x  —  sin  y. 

Suppose  x  to  increase  from  0  to  90°  ;  then  y  will  decrease  from 
180°  to  90°,  and  the  two  values  of  a  will  become  equal,  each  being 
4  (90°  rp  b).  But,  as  has  been  shown,  this  value  of  a  gives  a  maxi- 
mum for  r. 

175.  The  Greatest  Height  of  a  Projectile.—  To  find  the 
greatest  height  to  which  thfc  projectile  will  ascend,  it  must  be  con- 
sidered that  a  body  projected  perpendicularly  upward,  will  rise  to 
the  same  height  from  which  it  must  have  fallen  to  acquire  the 
velocity  of  projection  (Art.  24).  Since  v  represents  the  whole 
velocity  of  projection  in  an  oblique  direction,  and  since  a  is  the 
angle  of  elevation,  therefore  v  sin  a  is  that  component  of  the 
velocity  which  acts  directly  upward.  And  the  space  described 

v* 
vertically  by  tnis  component  of  the  velocity  is  [(3)  Artw28]  s  =  jr—  . 

"9 
Hence,  substituting  h  for  s,  and  v  sin  a  for  v,  we  have 


If,  therefore,  the  angle  of  elevation  and  the  velocity  of  projec- 
tion are  given,  the  greatest  height  is  found  as  above.  Or,  if  the 
angle  of  elevation,  and  that  of  the  plane,  be  given,  along  with  the 
range  (r)  or  the  time  (t),  then  let  v  be  found  first,  as  in  Art.  173  ; 
after  which  h  may  be  obtained  from  equation  (4). 

If  the  velocity  of  projection,  and  the  greatest  height  to  which 
the  projectile  rises,  were  given,  equation  (4)  will  determine  the 

^       .        ,       v*  sin2  a        .  „          2  a  h 
angle  of  elevation.    For  since  h  =  —  ~  -  ,  .*.  sin  a  =   —  ^—  ,  and 


sin  a  — 


176.  Particular  Formulae  for  a  Horizontal  Plane. — The 

preceding  equations  become  much  more  simple  when  the  projec- 
.  tion  is  above  a  horizontal  plane ;  for  then  6  =  0;  therefore  sin  b 
=  0,  and  cos  b  =  1;  hence,  from  equations  (1),  (2),  and  (3),  we 
have 

2  v  sin  a  ^  ,» 

t= cc  vsma (1), 

y 

=f£ en 


EQUATION  OF  THE    PATH   OF  A  PROJECTILE.    113 


FIG.  128. 


On  a  horizontal  plane,  therefore,  we  have  the  following  the- 
orems: 

I.  The  TIME  OF  FLIGHT  varies  as  the  velocity  of  projection  mi:l- 
tiplied  ly  the  sine  of  the  angle  of  elevation. 

II.  The  RANGE  varies  as  the  square  of  the  velocity  of  projection, 
multiplied  ly  the  sine  of  twice  the  angle  of  elevation. 

Moreover,  since  the  sine  of  twice  45°  equals  the  sine  of  90°, 
which  equals  radius,  hence,  by  Theorem  II, 

III.  The  RANGE  is  GREATEST  ivhen  the  angle  of  elevation  is 
45°,  and  is  the  same  at  elevations  equally  above  and  leloiv  45°. 

IV.  The  TIME  OF  FLIGHT  is  GREATEST  when  the  body  is 
thrown  perpendicularly  upward. 

177.  The  Equation  of  the  Path  of  a  Projectile.— Sup- 
pose the  body  is  projected  from  A  (Fig.  128)  in  the  direction  A  T 
with  a  velocity  v,  and  let 
A  X,  horizontal,  and  A  Y", 
vertical,  be  rectangular 
axes. 

The  components  of  v 
along  the  axes  are,  v  cos  a 
for  A  X,  and  v  sin  a  for 
A  Y.  At  the  end  of  the 
time  t,  suppose  the  body  to 
be  at  C.  Denote  the  co-or- 
dinates of  the  point  C by  x 
and  y',  then  x  =  t  v  cos  a, 
and  y  —  B  D  —  B  C  =tv  sma  — 

Eliminating  t,  we  have 

gx* 

y  —  x  tan  a  —  ^—^ — — > 
2  v*  cos2  a 

an  equation  expressing  the  relation  between  x  and  y  for  any  value 
of  t  whatever,  and  consequently  the  equation  of  the  path. 

To  find  the  range  A  E,  make  y  =  0;   then  x  =  0,  or  x  — 
%  vz  sin  a  cos  a 

g 

The  first  value  of  x  corresponds  to  the  point  A  ;  the  second  is 
the  range,  A  E. 

To  find  Nthe  time  of  flight,  make  x  =  r  in  the  equation  x  — 

t  v  cos  a,  and  we  have  t  =  - 


vcosa 


a  x* 
If  a  =  0,  y  =  —  5— j,  the  equation  of  the  path  when  the  body 

</  v  .    • 

is  thrown  horizontally.     Since  y  is  negative  for  all  values  of  x, 
every  point  of  the  path,  except  A,  lies  below  a  horizontal  line 


114  MECHANICS. 

drawn  through  the  point  of  projection.  A  G'  represents  the  path 
when  the  body  is  projected  in  the  line  A  X. 

If  positive  ordinates  are  estimated  from  A  X  downward,  the 

a  x* 
equation  may  he  written  y  =  |-^. 

178.  To  Find  the  Range  on  an  Oblique  Plane. — Let  b 

be  the  inclination  of  the  plane  to  the  horizon ;  then  y  —  ±  x  tan  b 
is  the  equation  of  the  line  in  which  the  oblique  plane  intersects 
the  plane  of  the  projectile's  path.  Combining  this  with  the  equa- 

a  x* 
tion  y  =  x  tan  a  —  H~T — —  >  we  nave   ±  &  tan  &  =  a  tan  a  — 

fy  V    COS    tt> 

a  x*  ,  A       -,         2  v*  cos2  a  ,,  ,       ,  x 

— — ;  whence  x  =  0,  and  x  =  -       (tan  a  =F  tan  b)  = 

2  v*  cos2  a '  g 

2^cosasin(ajFj)    and  hence  ^  ^  be  r  =     x      = 

#  cos  &  cos  & 
2  v2  cos  fl  sin  (a  =f  £) 

#  cos2  6 

179.  Questions  on  Projectiles. — 

1.  A  gun  was  fired  at  an  elevation  of  50°,  and  the  shot  struck 
the  ground  at  the  distance  of  4898  feet ;  with  what  velocity  did  it 
leave  the  gun,  and  how  long  was  it  in  the  air  ? 

Ans.  Velocity,  400  feet  per  second. 
Time,  19.05  seconds. 

2.  Eange  4898  feet,  time  of  flight  16  seconds;  required  the 
angle  of  elevation  and  the  velocity  of  projection  ? 

Ans.  a  =  40°  3',  v  =  400  feet  per  sec. 

3.  Range  2898  feet,  velocity  of  projection  389.1  feet,  what  were 
the  elevation  and  time  of  flight? 

Ans.  a  =  19°  or  71°,  t  =  7.87  or  22.86  sec. 

4.  Elevation  40°,  range  4898;   required  the  range  when  the 
elevation  is  29^°  Ans.  4263. 

5.  Elevation  40°  3',  time  of  flight  16  seconds;  required  the 
range  and  velocity  of  projection  ?      Ans.  r  =  4898,  v  =  400  ft. 

6.  Velocity  510  feet  per  sec.,  time  of  flight  15  seconds,  to  find 
the  elevation  and  range.  Ans.  a  —  28°  14',  r  =  6740. 

7.  On  a  slope  ascending  uniformly  above  a  horizontal  plane  at 
an  angle  of  10°  20;,  a  ball  was  fired  at  an  angle  of  elevation  above 
the  horizon  of  34°,  and  with  a  velocity  of  401  feet  per  second ; 
what  was  the  range  on  the  slope  when  the  gun  was  directed  up  the 
hill,  and  what  when  directed  downward  ? 

Ans.  3438  and  5985  feet. 

8.  What  will  be  the  time  of  flight  for  any  given  range,  the 

angle  of  elevation  being  45°  ?  .  /2~7. 

Ans.  t  =  V  — 


CENTRAL    FORCES.  115 

i 

9.  Having  given  the  angle  of  elevation,  to  determine  the  veloc- 
ity, so  that  the  projectile  may  pass  through  a  given  point. 

x'        /  <7 

Ans.  v  =  -     -  V  =-7—  — -,  where  x1  and  y'  are 

cos.  a  r   2  (x1  tan  a  —  y') 

the  co-ordinates  of  the  given  point. 

10.  Find  the  angle  of  elevation  and  velocity  of  projection  of  a 
shell,  so  that  it  may  pass  through  two  points,  the  co-ordinates  of 
the  first  being  x'  —  1700  ft.,  y1  —  10  ft,  and  of  the  second,  x"  = 
1800  ft.,  y"  =  10  ft  Ans.  a  =  39'  19",  v  =  2218.3  ft. 

CENTRAL  FORCES. 

180.  Central  Forces  Described. — Motion  in  a  curve  is 
always  the  effect  of  two  forces ;  one  an  impulse,  which  alone  would 
cause  uniform  motion  in  a  straight  line ;  the  other  a  continued 
force,  which  urges  the  body  toward  some  point  out  of  the  original 
line  of  motion.    The  first  is  called  the  projectile  force,  the  second 
the  centripetal  force. 

The  centripetal  force  may  be  resolved  into  two  components ; 
one  in  the  direction  of  the  tangent,  the  other  perpendicular  to  it. 
The  tangential  component  will  accelerate  or  retard  the  motion  in 
the  curve  according  as  it  acts  luith  the  projectile  force,  or  in  oppo- 
sition to  it.  When  the  body  moves  in  the  circumference  of  a 
circle,  the  tangential  component  of  the  centripetal  force  is  0,  and 
hence  the  motion  is  uniform. 

If  the  centripetal  force  should  cease  to  act  at  any  instant,  the 
body,  by  its  inertia,  would  immediately  begin  to  move  in  a  straight 
line  tangent  to  the  curve  at  the  point  where  the  body  was  when 
the  force  ceased  to  act. 

Since  the  body,  by  its  inertia,  tends  to  move  in  a  tangent,  there 
is  a  continued  outward  pressure  directed  from  the  centre  of  curva- 
ture ;  this  is  called  the  centrifugal  force.  In  circular  motion  it  is 
equal  to  the  centripetal  force,  and  directly  opposed  to  it. 

181.  Expressions  for  the  Centrifugal  Force  in  Circular 
Motion. — 

1.  Let  r  =  the  radius  of  the  circle,  v  = 
the  velocity  of  the  body,  c  =  the  distance 
through  which  the  centrifugal  force  causes 
the  body  to  move  in  one  second,  and  let 
A  B  (Fig.  129)  be  the  arc  described  in  the 
infinitely  small  time  t\  then  A  E  =  v  t, 
and,  by  a  method  similar  to  that  employed 
in  the  discussion  of  the  force  of  gravity,  it 
may  be  shown  that  B D  =  ct\ 


116  MECHANICS.  * 

But  A  B,  being  a  very  small  arc,  may  be  considered  as  equal 
to  its  chord,  which  is  a  mean  proportional  between  A  E  and  the 

v2  £* 
diameter  2  r.    Hence  c  f  =  -—  ,  or 


If  this  be  doubled,  then  (Art.  25)  --  is  the  velocity  which  the 

centrifugal  force  is  capable  of  generating  in  one  second,  and  this  is 
sometimes  taken  as  the  measure  of  the  centrifugal  force. 

From  (1)  it  follows  that  in  equal  circles  the  centrifugal  force 
varies  as  the  square  of  the  velocity. 

2.  The  value  of  c  may  be  expressed  in  a  different  form.  Let 
I*  =  the  time  of  a  complete  revolution  ;  then  %Trr  =  vt';  whence 

v  =  —T~*    This  substituted  in  (1)  gives 


9) 


Hence  the  centrifugal  force  varies  directly  as  the  radius  of  the 
circle,  and  inversely  as  the  square  of  the  time  of  revolution. 

3.  Let  w  =  the  weight  of  the  revolving  body,  and  c'  =  the 
centrifugal  force  expressed  in  pounds  ;  then 


,  ,  .  ,  . 

^v:c::^  a  :•=--)  whence  c  —  ---  .    .     (3) 
'   2  ry  rg 

Let  n  =  the  number  of  revolutions  per  second  ;  then 
v  =  2  TT  r  n,  and  (3)  becomes 

47Ta 

c  =  --  .w  .r  .n     .......     (4) 

182.  Two  Bodies  Revolving  about  their  Centre  of 
Gravity.  —  Let  A  and  B  (Fig.  130)  be  two  bodies  connected  by 
a  rod,  and  let  them  be  made  to  ^    _ 

revolve  about  the  centre  of 
gravity  (7;  then  by  (4)  the 
centrifugal  force  of  A  will  be 

-—.A.AC.  n\  and  of  B,  —  .  B  .  B  C.n\ 
9  9 

But  C  being  the  centre  of  gravity  of  the  two  bodies,  A  .  A  C  = 
B..B  C;  .*.  the  centrifugal  force  of  A  equals  that  of  B.  Hence 
if  two  bodies  revolve  in  the  same  time  about  an  axis  passing  through 
their  centre  of  gravity,  there  will  be  no  strain  upon  that  axis. 

183.  Centrifugal  Force  on  the  Earth's  Surface.—  As  the 

earth  revolves  upon  its  axis,  all  free  particles  upon  it  are  influenced 
by  the  centrifugal  force.    Let  N  8  (Fig.  131)  be  the  axis,  and  A  a 


EXAMPLES    ON    CENTRAL    FORCES. 


117 


particle  describing  a  circumference  with  the  radius  A  0.    Put  r  = 

C  Q,r'  =  A  0,1  =  the  angled  C  Q, 

the  latitude,  c  =  the- centrifugal  force 

at  the  equator,  c1  =  the  centrifugal 

force  at  A,  v  =  velocity  of  Q,  and  v' 

—  velocity  of  A ;  then 


But  v  :  v' 


r :  r 


whence 


v  r' 


Again,  from  the  triangle  AGO 

we  have  r'  =  r  cos  /;    hence  v'  = 

v2  cos2 1       v*  cos  I 
v  cos  I,  and  c'  —  ^—     — -.  —  — 
2  r  cos  I 


0 
2  r 


Comparing  this  value  of  c' 


with  that  of  c,  we  have 

c'  =  c  cos  £. 

That  is,  £Ae  centrifugal  force  at  any  point  on  the  earth's  surface  is 
equal  to  the  centrifugal  force  at  the  equator,  multiplied  fiy  the  cosine 
of  the  latitude  of  the  place. 

Let  A  B  represent  the  centrifugal  force  at  A,  and  resolve  it 
into  A  D  on  C  A  produced,  and  A  F,  tangent  to  the  meridian 
N  Q  S  ;  then,  since  the  angle  D  A  B  —  A  C  Q  =  I,  we  have 

A  D  =  A  B  cos  I  =  c  cos  I  .  cos  I  =  c  cos2  /. 

That  is,  that  component  of  the  centrifugal  force  at  any  point  which 
opposes  the  force  of  gravity  is  equal  to  the  centrifugal  force  at  the 
equator,  multiplied  ~by  the  square  of  the  cosine  of  the  latitude  of 
the  place. 

In  like  manner  we  find  A  F  —  A  B  sin  I  =  c  cos  I  sin  I  = 

—  .    From  this  equation  we  see  that  the  tangential  com- 

ponent is  0  at  the  equator,  increases  till  I  =  45°,  where  it  is  a 
maximum  ;  then  goes  on  diminishing  till  I  =  90°,  when  it  again 
becomes  0. 

The  effect  of  A  D  is  to  diminish  the  weight  of  the  particle, 
while  the  effect  of  A  JFis  to  urge  it  toward  the  equator. 

184.  Examples  on  Central  Forces.  — 

1.  A  ball  weighing  10  Ibs.  is  whirled  around  in  a  circumference 
of  10  feet  radius,  with  a  velocity  of  30  feet  per  second.    What  is 
the  tension  upon  the  cord  which  restrains  the  ball  ? 

Ans.  28  Ibs.  nearly. 

2.  With  what  velocity  must  a  body  revolve  in  a  circumference 
of  5  feet  radius,  in  order  that  the  centrifugal  force  may  equal  tho 
weight  of  the  body  ?  Ans.  v  =  12.7  ft. 


118  MECHANICS. 

3.  A  ball  weighing  2  Ibs.  is  whirled  round  by  a  sling  3  feet  long, 
making  4  revolutions  per  second.    What  is  its  centrifugal  force  ? 

Ans.  117.84  Ibs. 

4.  A  weight  of  5  Ibs.  is  attached  to  the  end  of  a  cord  3  feet  long 
just  capable  of  sustaining  a  weight  of  100  Ibs.    How  many  revo- 
lutions per  second  must  the  body  make  in  order  that  the  cord 
may  be  upon  the  point  of  breaking  ?  Ans.  n  =  2.3  nearly. 

5.  A  railway  carriage,  weighing  7  tons,  moving  at  the  rate  of 
30  miles  per  hour,  describes  an  arc  whose  radius  is  400  yards. 
What  is  the  outward  pressure  upon  the  track  ?     Ans.  786  +  Ibs. 

6.  A  hemisphere  has  its'  base  fixed  in  a  horizontal  position, 
and  a  body,  under  the  influence  of  gravity,  moves  down  the  con- 
vex side  of  it  from  the  highest  point.    How  far  from  the  base  will 
the  body  be  when  it  leaves  the  surface  of  the  hemisphere  ? 

Ans.  |  r. 

185.  Composition  of  two  Rotary  Motions. — 
When  a  lody  is  rotating  on  an  axis,  and  a  force  is  applied  which 
alone  would  cause  it  to  rotate  on  some  other  axis,  the  body  will  com- 
mence rotation  on  an  axis  lying  between  them,  and  the  velocities  of 
rotation  on  the  three  axes  are  such,  that  each  may  ~be  represented  ~by 
the  sine  of  the  angle  between  the  other  two. 

Suppose  that  the  body  H  K  (Fig.  132)  is  rotating  on  A  JB, 
and  that  a  force  is  applied  to  make  it 
rotate  on  0  D.    Let  these  axes  intersect  FlG- 

within  the  body,  and  call  the  point  of 
intersection,  G.  Imagine  a  perpendicu- 
lar to  the  plane  of  the  axes  to  be  drawn 
through  Gj  and  let  P  be  a  particle  of 
the  body  in  this  perpendicular.  Sup- 
pose the  particle  P,  in  an  infinitely  small 
time  t,  to  pass  over  P  a  by  the  first  rota- 
tion, and  P  c  by  the  second.  Then, 
since  the  particle  will  describe  the  diago- 
nal P  e  in  the  time  t,  this  line  must  in- 
dicate the  direction  and  velocity  of  the 

resultant  rotation.  Therefore,  if  E  F  be  drawn  through  G,  per- 
pendicular to  the  plane  G  P  e,  E  F  is  the  axis  on  which  the  body 
revolves  in  consequence  of  the  two  rotations  given  to  it.  Since 
P  Gis  perpendicular  to  the  plane  A  G  C,  and  also  to  the  line  E  F, 
therefore  E  F  is  in  that  plane ;  that  is,  the  new  axis  of  rotation  is 
in  the  plane  of  the  other  two  axes.  The  angles  AGE  and  E  G  C, 
are  respectively  equal  to  the  angles  a  P  e  and  e  P  c,  the  inclina- 
tions of  the  planes  of  rotation.  But  the  lines,  Pa,  PC,  P  e, 
represent  the  velocities  in  those  directions  respectively;  and 
(Art.  41)  P  a  :  P  c  :  P  e  : :  sin  c  P  e  :  sin  a  P  e  :  sin  a  P  c;  there- 


THE    GYROSCOPE. 


119 


fore  P  a  :  P  c  :  P  e  : :  sin  C  G  E  :  sin  A  O  E  :  sin  A  O  (7;  or, 
the  velocities  on  the  three  axes,  (namely,  the  axes  of  the  compo- 
nent rotations,  and  of  the  resultant  rotation,)  are  such,  that  each 
may  be  represented  by  the  sine  of  the  angle  between  the  other 
two  axes. 

186.  The  Gyroscope. — The  gyroscope  affords  an  illustration 
of  the  composition  of  two  rotations  imparted  to  a  body.  As 
usually  constructed,  it  consists  of  a  heavy  wheel  G  H  (Fig.  133), 
accurately  balanced  on 

the  axis  a  b,  which  runs  FlG- 133- 

with  as  little  friction  as 
possible  upon  pivots  in 
a  metallic  ring.  In  the 
direction  of  the  axis, 
there  is  a  projection  B 
from  the  ring,  having  a 
socket  sunk  into  it  on 
the  under  side,  so  that  it 
may  rest  on  the  pointed 
standard,  S,  without 
danger  of  slipping  off. 

The  wheel  is  made 
to  rotate  swiftly  by  draw- 
ing off  a  cord  wound 
upon  a  b,  and  then  the 
socket  in  B  is  placed  on 
the  standard,  and  the  whole  left  to  itself.  Immediately,  instead 
of  falling,  the  ring  and  wheel  commence  a  slow  revolution  in  a 
horizontal  plane  around  the  standard,  the  point  A  following  the 
circumference  A  E  F,  in  a  direction  contrary  to  the  motion  of  the 
top  of  the  wheeL 

This  revolution  is  explained  by  applying  the  principle  of  com- 
position of  rotations  given  in  the  preceding  article.  The  particles 
of  the  wheel  are  rotating  about  the  horizontal  axis  a  b  by  the  force 
imparted  by  the  string.  The  force  of  gravity  tends  to  make  it 
fall,  that  is,  to  revolve  in  a  vertical  circle  around  the  axis  C  D  at 
right  angles  to  a  b.  Hence,  in  a  moment  after  dropping  the  ring, 
the  system  will  be  found  revolving  on  an  axis  which  lies  in  the 
direction  E  B,  between  A  B  and  C  D,  the  other  two  axes.  Now, 
gravity  bears  it  down  around  a  new  axis  perpendicular  to  E  B. 
Therefore,  as  before,  it  changes  to  still  another  axis  F B,  and  thus 
continues  to  go  round  in  a  horizontal  circle. 

The  only  way  possible  for  it  to  rotate  on  an  axis  in  a  new  posi- 
tion, is  to  turn  its  present  axis  of  rotation  into  that  position. 


120  MECHANICS. 

Hence,  the  whole  instrument  turns  about,  in  order  that  its  axis 
may  take  these  successive  positions. 

The  change  of  axis  is  seen  also  by  observing  the  resultant  of 
the  motions  of  the  particles  at  the  top  and  bottom  of  the  wheel. 
For  example,  G  is  moving  swiftly  in  the  direction  m  by  the  rota- 
tion around  a  ~b ;  by  gravity  it  tends  to  move  slowly  in  the  line  r, 
tangent  to  a  vertical  circle  about  the  centre  B.  The  resultant  is 
in  the  line  n,  tangent  to  the  wheel  when  its  axis  a  b  has  taken  the 
new  position  E  B. 

The  centre  of  gravity  of  the  ring  and  wheel  tends  to  remain  at 
rest,  while  the  resultant  of  the  two  rotations  carries  around  it  all 
other  parts,  standard  included,  in  horizontal  circles.  But  the 
standard  by  its  inertia  and  friction  resists  this  effort,  and  the  reac- 
tion causes  the  ring  and  wheel  to  go  around  the  standard. 


CHAPTER    IX. 

STRENGTH    OF    MATERIALS. 

187.  Longitudinal  Strength. — Strength  is  the  power  to  resist 
fracture ;  stress,  the  power  to  produce  fracture.    When  a  force  is 
applied  to  a  bar  or  rod,  to  pull  it  asunder,  its  strength,  called  in 
this  case  longitudinal  strength,  is  proportional  to  the  area  of  its 
cross-section.    Each  line  of  particles  in  the  direction  of  the  length 
of  the  bar  has  its  separate  strength,  and  the  whole  strength  there- 
fore depends  on  their  number,  that  is,  on  the  area  of  the  cross- 
section.    The/orm  of  the  cross-section  is  immaterial. 

188.  Lateral  Strength,  Support  at  Each  End.— When  a 

beam  rests  horizontally,  supported  at  both  ends,  and  pressed  by- a 

weight  at  the  centre,  its  strength  at  that  point  varies  as  the  area 

of  the  cross-section,  multiplied  by  the  depth  of  the  centre  of  gravity. 

Let  A  B  C  D  (Fig.  134)  represent  a  longitudinal  section  of  a 

FIG.  134. 


a 

tjf^^  ° 

\ 

b 

£L    * 

c 

m7      e'e 

^ 

B 

d                                     it 

/ 

i 

A 

L 

prismatic  beam  and  E  Tc  d  I  a  section  of  any  form  whatever,  at 
right  angles  to  the  axis  of  the  beam.  Let  G  be  the  centre  of 
gravity  of  the  cross-section,  and  g  h,  k  I,  m  n,  &c.,  the  width  of 


STRENGTH    OF    MATERIALS.. 


horizontal  laminae  of  the  beam.  Suppose  this  beam  to  be  sup- 
ported at  its  two  ends,  and  that  on  the  middle  of  it  at  E  there  is 
placed  a  weight,  W.  From  E  draw  E  d  perpendicular  to  the 
horizon,  and  cutting  the  several  laminse  in  the  points  a,  b,  c,  d,  &c. 

The  pressure  of  the  weight  W  tends  to  produce  a  fracture  in 
the  beam,  beginning  at  d,  and  passing  through  the  laminae  in 
succession  until  it  arrives  at  E. 

The  tendency  of  the  beam  to  resist  fracture  depends  partly 
upon  the  cohesion  of  its  corresponding  particles,  and  partly  upon 
the  distance  from  E  at  which  the  force  of  cohesion  acts.  E  may 
then  be  considered  as  the  centre  of  motion  of  a  lever,  at  the  ex- 
tremities of  whose  arms  E  d,  E  c,  E  b,  E  a,  &c.,  this  force  is  applied. 
Let  s  —  the  cohesive  force  of  one  line  of  particles,  then  s  x  g  h 
will  represent  the  strength,  of  the  lamina  whose  width  is  g  h, 
and  s  x  g  h  x  E  a  will  represent  the  power  of  this  lamina  to 
resist  fracture ;  hence  the  whole  power  of  the  beam  to  resist  frac- 
ture will  be 

s  (g  h  x  E a  +  Tel  x  E  b  +  mn  x  E c  +  &c.) 

Put  A  =  the  area  of  the  cross-section  and  G  =  the  depth  of 
the  centre  of  gravity  below  E ;  then  (Art.  78) 

_g  h  x  E  a  +  Icl  x  E  b  +  m  n  x  E  c  +  &c. 

or  A  .  G  =  g  h  x  E  a  +  Jcl  x  E  b  +  mn  x  EC  +  &c.,  and  hence 
the  strength  of  the  beam  is  equal  to  s  .  A  .  G  oc  A  .  G. 

189.  Special  Cases. — This  proposition  is  general,  and  ap- 
plies to  a  number  of  distinct  cases.  In  cylindrical  and  square 
beams,  since  the  area  of  the  section  varies  as  the  square  of  its 
depth,  and  the  distance  of  the  centre  of  gravity  from  the  point  E 
varies  as  the  depth,  their  strength  is  as  the  cube  of  the  depth.  In 
beams  whose  cross-section  is  a  rect- 
angle (Fig.  135),  the  strength  varies  as 
the  breadth  and  square  of  the  depth  ; 
for  here  the  area  being  as  the  product 
of  the  two  sides,  and  the  distance  of 
the  centre  of  gravity  from  E  being 
equal  to  half  the  vertical  side, 
and  therefore  proportioned  to  that 
side,  the  proposition  is,  that  the 
strength  varies  as  the  breadth  x  depth 

x  depth,  or  as  the  breadth  into  the  square  of  the  depth.  Hence, 
the  same  beam  with  its  narrow  side  upward,  is  as  much  stronger 
than  with  its  broad  side  upward,  as  the  depth  exceeds  the  breadth, 


FIG.  135. 


122 


MECHANICS. 


For  the  area  being  the  same  in  both  cases,  the  strengths  are  pro- 
portioned to  E  G  and  Eg,  or  as  A  B  to  A  C.  Thus  if  a  joist  be 
10  inches  broad  and  2^  thick,  it  will  bear  four  times  as  much 
weight  when  laid  on  its  edge  as  when  laid  on  its  side.  Hence  the 
modern  mode  of  flooring  with  thin  but  deep  pieces  of  timber. 

Again,  a  triangular  beam  is  twice  as  strong  when  resting  on  a 
side,  as  when  resting  on  an  edge.  For,  the  area  being  the  same  in 
both  cases,  the  strength 

varies  as  E  G  and  E  g  FlG-  186. 

(Fig.  136),  which  are  as 
2  to  1  (Art.  72).  These 
principles  apply  not  only 
to  beams,  but  to  bars,  and 
similar  forms  of  every  sort 
of  matter. 

190.  Stress  from  the  Weight  of  the  Beam.  —  The  stress 
arising  from  the  weight  of  a  beam  varies  as  the  product  of  the 
length  and  weight    Let  L  =  length,  and  W  =  weight  of  the  whole 
beam,  w  =  weight  of  the 
portion  A  C  (Fig.  137).  FlG  137- 

The  pressure  on  each 
prop  is  -^  W]  this,  there- 
fore is  the  force  acting  at 
A  which  tends  to  fracture  the  beam  at  C  with  an  energy  expressed 
by  $  W  x  A  G. 

Now  suppose  the  portion  C  B  to  be  held  firmly  in  solid  ma- 
sonry, and  a  force  —  4  W  to  act  upward  at  A,  and  another  =  w  at 
the  middle  of  A  (7  to  act  downward,  the  tendency  to  produce  frac- 
ture will  be  the  same  as  before,  and  hence  the  stress  at  C  will  be 
J  WxA  G-wx$AC=$(W-w)AC.  But  AB\  A  C\\  W:w\ 

A  G 
whence  w  =  —r-^  W',  so  that  the  expression  for  the  stress  will  be 


A 


-  A 


A  B  being  constant,  the  stress  at  any  point  G  varies  as  the  rectan- 
gle of  the  two  lines  A  C  and  B  C,  and  is  greatest  when  A  G  =  B  C, 
or  when  G  is  at  the  middle  of  the  beam,  where  the  stress  is 


*•  A    "B  *  T  8' 

If,  therefore,  we  use  the  term  relative  strength  to  denote  the  ratio 
of  the  strength  to  the  stress,  and  represent  this  ratio  by  8,  we 
shall  have 

<?     A'& 
L.  W 


STRENGTH    OF    MATERIALS.  123 

If  beams  are  similar)  then  the  above  ratio  is  inversely  as  any 
one  of  their  three  dimensions.  Being  similar,  their  length,  breadth, 
and  thickness  are  proportional.  Let  D  represent  any  one  of  these 
three  dimensions.  Then, 

Since  A  oc  Z)2  and  G  oc  D, 

A  .  G  x  Z)3;  also,  L  oc  D,  and  W oc  Z>3; 


Hence  the  relative  strength  of  large  structures  is  less  than  that 
of  smaller  similar  ones.  If  a  model  is  three  feet  long,  and  the 
structure  75  feet  long,  then  the  structure  is  25  times  weaker  rela- 
tively than  the  model. 

191.  Additional  Weight  at  the  Centre.— If  a  weight  W 
is  uniformly  distributed  through  a  beam  whose  length  is  L,  the 
stress  arising  from  the  weight  is  (Art.  190)  £  L  .  W.  If  the  same 
weight  is  placed  at  the  middle  of  the  beam,  the  stress  is  J  L  .  -£  W 
=  J  L  W ',  hence,  if  a  weight  is  placed  at  the  middle  point,  the 
stress  is  twice  as  great  as  when  distributed  uniformly  through  the 
beam. 

If  W  is  the  weight  of  the  beam,  and  a  weight  W  is  placed  at 
the  middle,  then,  from  what  has  just  been  shown,  the  relative 
strength  will  be 

A.G  A.G 


If  the  beam  is  small  compared  with  the  weight  laid  upon  it, 
then 

A.  G 
C  L  .  W 

In  order  that  the  foregoing  general  formulas  may  be  applied  to 
practice,  so  as  to  find  the  actual  strength  of  bars  or  beams,  it  is 
necessary  to  have  some  standard  of  strength  ascertained  by  experi- 
ment, which  may  be  employed  as  the  unit  of  comparison.  For 
example,  it  is  found  by  experiment  that  a  stick  of  oak,  one  foot 
long  and  one  inch  square,  is  able,  when  supported  at  both  ends,  to 
sustain  a  weight  of  600  pounds ;  and  that  a  bar  of  iron  of  the  same 
dimensions  would  sustain,  in  the  same  circumstances,  2190  pounds. 
The  oak  weighs  half  a  pound,  and  the  iron  three  pounds.  With 
these  data  applied  to  the  foregoing  formulas,  we  may  solve  such 
problems  as  the  following: 

1.  What  weight  can  be  sustained  at  the  middle  point  of  a  pris- 
matic beam  of  oak,  whose  length  is  6  feet,  and  its  end  4  inches 
square  ? 


124  MECHANICS. 

If  the  weight  of  the  bar  is  left  out  of  account, 

A.  G       I2 .   4  ,  42 .   2     „     , , 

-r  =  3 — ^-,  for  one  case ;  and  ^ — =,  for  the  other. 


L.W  ~  1  .  600'  A  6  .  If ' 

But  these  expressions  are  to  be  equal  at  the  moment  of  rupture  of 
both  beams,  since  at  that  moment  they  have  the  same  relative 
strength ; 

,.^^,,...W'  =  6400  Ibs.    Ans. 
If  the  weight  of  the  beams  be  considered,  then 


In  this  example,  the  weight  of  the  large  beam  is  known  to  be 
48  Ibs.,  from  the  given  dimensions  and  weight  of  the  small  one. 

2.  What  must  be  the  depth  of  a  beam  in  the  form  of  a  rectan- 
gular prism,  whose  breadth  is  2  inches  and  length  8  feet,  to  sup- 
port a  weight  of  6400  pounds,  its  own  weight  not  being  taken  into 
consideration  ? 

IV.   4        2.  D.i  D        _      nKO  . 

--  -   Ans' 


3.  What  weight  can  be  supported  at  the  middle  point  of  a  bar 
of  iron  10  feet  long,  and  the  side  of  whose  square  end  is  3  inches, 
its  own  weight  not  being  taken  into  consideration  ? 

Ans.  5913  pounds. 

192.  Additional  Weight,  at  Any  Point. — The  stress  pro- 
duced by  a  weight,  at  any  point,  is  as  the  product  of  the  two  dis- 
tances from  the  ends.    In  Fig.  138,  ac- 
cording to    the   theorems    for    parallel  FIG.  138. 

W  x  B  G 
forces,  pressure  at  A  =  — -j-= — ,  and 

W  x  A  C 
pressure  at  B  =  j-= — .    But  the  re- 

u!±  JO 

action  of  either  point  of  support  is  equal  ^  w 

to  the  pressure  on  that  point;   and  this  force  acts  at  C  with  a 

leverage  A  G  on  one  side,  and  B  G  on  the  other,  so  that  the  stress 

W  x  B  G  W  x  AG 

at  C  = -po —  x  A  G,  or  — j~= —  x  B.  G,  either  of  which 

.0.  Jj  *A.  jj 

expressions  —  stress  at  C,  and  oc  A  G  x  B  C.  And  since  this 
rectangle  is  greatest  when  A  G  =  C  B,  and  diminishes  as  these 
lines  become  more  and  more  unequal  in  length,  so  the  tendency 
of  a  horizontal  bar  to  break  is  greatest  in  the  middle,  and  decreases 
toward  the  points  of  support. 

193.  Form  f  ~r  Equal  Strength. — Hence  a  beam,  in  order 
to  be  equally  strong  throughout,  must  be  thickest  in  the  middle ; 


STRENGTH    OF    MATERIALS. 


125 


and  if  the  sides  of  such  a  beam  are  parallel  planes,  the  figure  of  the 
beam  must  be  elliptical. 

For  let  the  curve  A  P  D  M  (Fig.  139),  whose  axis  is  A  D,  rep- 
resent a  longitudinal  section, 
and  let  a  =  the  thickness  or 
breadth  of  the  beam  ;  then  a 
section  of  the  beam  perpen- 
dicular to  the  axis  at  any  point 
G  will  be  a  rectangle,  whose 
breadth  is  a,  depth  P  M,  and 
the  depth  of  its  centre  of  gravity  A  P  M.  Hence,  the  tendency  of 
the  beam  to  resist  fracture  at  any  point  C  is  as  a  x  P  M2  ;  but 
the  stress  at  C  is  as  A  C  x  0  D  ;  therefore 

a  x  P  M*  P  M* 

the  relative.strength  at  G 


hence,  if  P  M*  cc  A  C  x  C  D,  the  strength  will  be  the  same  at 
every  point  :  but  in  this  case  the  curve  A  P  D  M  is  an  ellipse, 
whose  major  and  minor  axes  are  A  D  and  F  /£ 

Therefore,  in  the  use  of  horizontal  rectangular  timbers  in 
building,  much  of  the  timber  is  useless,  though  it  would  only  be  a 
waste  of  labor  to  remove  the  redundant  parts.  But  iron  beams 
are  often  made  of  curved  forms,  which  combine  strength  with 
lightness  and  economy  of  material.  The  convex  form  has  some- 
times been  adopted  in  the  iron  bars  for  railroad  tracks,  as  shown 

in  Fig.  140. 

FIG.  140. 


194.  Lateral  Strength,  Support  at  One  End. — When  a 
prismatic  beam  is  secured  firmly  in 
a  wall  at  one  end,  the  same  general 
statements  hold  true  as  in  relation 
to  beams  supported  at  both  ends. 

1.  The  strength  varies  as  the 
area  of  the  cross-section  multiplied 
ly  the  height  of  the  centre  of  gravity. 

-LziABE  F,  a  I  ef  (Fig.  141), 
represent  the  longitudinal  sections 
of  two  prismatic  beams  fixed  hori- 
zontally into  the  wall  H  K  L  M ; 
then  the  tendency  of  these  beams  to 
resist  fracture  at  the  ends  E  F,  e  /, 


126 


MECHANICS. 


where  they  are  inserted  into  the  wall,  will  be  measured  ly  the  area 
of  the  cross-section  into  the  height  of  its  centre  of  gravity  ;  for  in 
this  case  the  fracture  will  begin  at  the  upper  points  F9  /,  and  end 
at  the  lower  points  E,  e ;  that  is,  strength  oc  A  .  G. 

But  the  tendency  to  produce  fracture  will  be  the  weight  of  the 
beams,  acting  at  the  distance  of  their  centres  of  gravity  from  the 
ends  E  F,  ef.  Hence, 

A.G 


2.  If  an  additional  weight  lies  on  the  end  of  the  beam,  then 

<?  A-G 

°  L  Q  W+  W')' 

3.  If  beams  are  similar,  S  oc  yp 

If  any  other  cross-section  be  taken,  as  C  D,  and  W  represent 
the  weight  from  it  to  the  end  A  B,  then,  by  the  same  course  of 

A    G 
reasoning  as  before,  the  relative  strength  at  (7 />(==$)  oc       *       . 

./a  (j  .  I  \ 

If  W  represent  a  weight  at  the  end,  the  strength  to  support 

A.  G 


W,  at  any  section  C  D,  is  as 

A.  G 

A  C' 


A  0.  W 


7 ;  and  if  W  is  constant, 


FIG.  142. 


W 


195.  Forms  of  Equal  Strength  at  Every  Section. — 

1.  A  beam  supported  at  one  end,  and  having  the  form  of  a 
wedge,  whose  triangular  sides  are 

parallel  to  the  horizon,  has  equal 
strength  at  every  section,  for  sup- 
porting a  weight  at  the  extremity. 
Let  the  wedge,  in  Fig.  142,  have  the  . , 
uniform  depth  d\  then  G  —  \  d\ 
and  the  area  of  the  section  at  G  is 

E  D  oc  A  C\    hence  the  relative 

A  C 
strength  is  as  -4-77;  that  is,  it  is 

constant. 

2.  A  beam,  whose  vertical  sides 
are  parallel,  and  whose  longitudinal 
section,  parallel  to  the  sides,  is  a 
semi-parabola,  has  equal  strength  at 
every  section.    In  Fig.  142,  let  d  = 

the  uniform  thickness;  then  A  =  d  .  G'  D'9  G  =  J  G' D' \  .-.  A  .  G 


STRENGTH    OF    MATERIALS.  127 


J  d  .  C'  D"*  oc  a  D'\    Hence,  8  «    jf~  ;  but  A'C><x.  C'  Dn  ; 

^3 

icn  is  constant. 


196.  Prismatic  Beams  Breaking  by  their  own  Weight- 

Suppose  beams  of  prismatic  or  cylindrical  form  to  have  similar 
cross-sections,  one  dimension  of  which  is  D,  and  to  be  supported 

at  both  ends,  or  only  at  one  end  ;  then  8  cc  '  ---  —  —  x 

D3 
L  ('   W  -4-  W'Y    ~^"S  ^ariation,  thrown  into  the  form  of  a  full 

proportion,  becomes  8:,::  L(^'+W>)  '  ifi/f^)'  For  ex" 
ample,  let  the  beams  be  cylinders,  whose  lengths  are  L,  ?;  their 
diameters,  Z>,  t?;  weights,  TF,  w;  while  W,  w',  are  additional 
weights  laid  on  their  middle  points,  or  the  unsupported  ends. 
Then  the  above  proportion  gives  their  relative  strength.  Now 
let  the  second  beam  have  no  weight  laid  upon  it  ;  that  is,  let 


hence  Sis::  ^.  '       :  -=—  —  ;  which  expresses  their  relative 

strength,  when  the  diameters  are  equal,  and  the  second  beam 
is  not  loaded.      If  their  lengths   and  weights  become  such  as 

to  cause  the  beams  to  break,  then,  since  .S  =  s,  .'.  -f-rr-^r  — 

J-j  (^-  \\  -f- 

W+2W'\-\ 

jp  —  f  ;  w 

gives  the  length  of  a  beam  that  breaks  by  its  own  weight. 

Let  the  prismatic  or  cylindrical   beams  be  similar  to  each 

other;  then  Z>3  :  d9  :  :  L3  :  V  ;   .:  S  :  s  :  :  -,-^-w-l  :  —-.   But 

±W+  }V  '    £  w 

the  weights  of  similar  solids  of  the  same  density  are  as  the  cubes 

w  r 

of  their  homologous  dimensions;  /.  w  :  W:  :  I3  :  L3;  .:  w  =      '     ; 

L 

L9          2  L3 

hence,  by  substitution,  S  :  s  :  :  ,  ^+  }yt  :  -^-  -T     And  when  the 

T2  n    T3 

beams  break,  since  S  =  s,  therefore  -p^  —  JTT,  =  -7^-7  ;  .*•  J  TF+  TT'' 
—  "o~F"5  or  Z  =  -      —    --  -,     If,  therefore,  a  cylindrical  beam 


128  MECHANICS. 

whose  length  is  L  breaks  with  the  given  weight  IF'  placed  upon 

it,  a  similar  cylindrical  beam  whose  length  is  — - — -== •  will 

break  with  its  own  weight. 

The  same  reasoning  is  applicable  to  a  beam  of  any  prismatic 
form,  whatever  the  shape  of  the  cross-section. 

197.  Comparison  of  Beams  Supported  at  One  End 
and  at  Both  Ends.— If  a  horizontal  beam  be  supported  at  both 
ends,  the  stress  produced  by  its  own  weight,  IF,  is  measured  by 
IL  x  W  (Art.  190). 

If  the  beam  be  supported  at  one,  end  only,  the  stress  is  measured 
by  the  whole  weight  applied  at  the  centre  of  gravity,  and  conse- 
quently the  stress  =  J  L  x  W. 

Therefore  a  beam  supported  at  both  ends  has  four  times  the 
relative  strength  of  the  same  beam  supported  only  at  one  end. 
And  if  a  certain  beam  supported  at  one  end  breaks  by  its  own 
weight,  a  beam  of  the  same  dimensions  twice  as  long  will  break 
by  its  own  weight  when  resting  on  two  supports. 

If,  however,  instead  of  the  weight  of  the  beam  itself,  this  is  left 
out  of  the  account,  and  a  weight  W  be  added,  then  the  stress  on 
the  beam  when  supported  at  one  end  will  be  measured  by  L  x  IF'; 
while,  in  the  case  of  the  beam  supported  at  both  ends,  the  weight 
being  at  the  middle  point  of  the  beam,  the  stress  is  measured  as 
before,  by  }  L  x  W  (Art.  191).  Therefore,  a  weight  placed  at  the 
end  of  a  beam  supported  only  at  one  end  produces  four  times  the 
stress  as  the  same  weight  placed  at  the  middle  of  the  beam  when 
supported  at  both  ends. 

1.  What  must  be  the-  length  of  a  beam  of  oak  one  inch  square, 
supported  at  both  ends,  which  is  just  capable  of  bearing  its  own 
weight  ? 

By  Art.  191,  a  beam  of  oak  1  foot  long  and  1  inch  square, 
weighing  |  pound,  just  supports  600  pounds.  And  by  Art.  196, 

,  (w+2  WY * 

the  expression  I  =  L  I ^ 1   denotes  that  when  a  beam 

whose  length  is  L  breaks  when  W  is  placed  upon  it,  I  is  the  length 
of  a  beam  that  will  break  with  its  own  weight ;  consequently,  since 


here  L  =  1,  W  =  £,  and  IF'  =  600,  I  =  (*  +  J120Q ^  (3401)* 
=  49  feet 

2.  Two  beams  are  of  equal  length  and  weight,  the  first  being 
a  square  prism  whose  section  is  4  inches  square,  the  second  a  rect- 
angular prism,  8  by  2  inches ;  how  much  stronger  is  the  second 


PROBLEMS    IN    MECHANICS.  1^9 

beam  than  the  first,  and  how  much  stronger  when  laid  on  the  nar- 
row than  on  the  broad  side  ?  (Art.  189.) 

Ans.  The  second  beam  is  twice  as  strong  as  the  first,  and 
four  times  as  strong  when  laid  on  the  narrow  as 
on  the  broad  side. 

198.  Structures     Relatively    Weaker    as    they    are 
Larger. — The  foregoing  articles  explain  the  observed  fact  that  the 
relative  strength  of  every  kind  of  structure  becomes  less  as  its  size 
is  increased.    For,  the  absolute  strength  increases  as  the  square 
of  one  of  the  dimensions,  while  the  weight  increases  as  the  cube  of 
the  same.    A  model,  therefore,  has  far  greater  relative  strength 
than  the  building  copied  from  it ;  and  in  respect  to  every  kind  of 
structure,  there  are  limits  of  magnitude  which  cannot  be  exceeded. 

The  same  fact  is  observed  in  the  animal  and  vegetable  world. 
Relatively  to  size,  insects  are  very  much  stronger  than  large  ani- 
mals, and  shrubs  stronger  than  trees. 

199.  Strength  cf  Solid  and  Hollow  Cylinders.— If  a 

solid  and  a  hollow  cylinder,  of  equal  length,  have  the  same  quan- 
tity of  matter,  so  that  the  area  of  their  cross-sections  shall  be 
equal,  then  their  strength  will  be  in  the  ratio  cf  the  distances  of 
their  centres  of  gravity  from  the  upper  surfaces.  But  the  centres 
of  gravity  being  at  the  centres  of  the  cross-sections,  it  follows  that 
the  strength  of  the  solid  cylinder  will  be  less  than  that  of  the  hol- 
low cylinder  in  the  ratio  of  the  diameter  of  the  former  to  that  of 
the  latter. 

It  appears,  therefore,  that  the  strength  of  a  tube  is  always 
greater  than  the  strength  of  the  same  quantity  of  matter  made 
into  a  solid  rod  of  the  same  length ;  and  leaving  out  of  view  the 
diminished  rigidity,  there  would  seem  to  be  no  limit  to  the  strength 
which  might  be  given  to  such  a  cylinder  by  increasing  its  diameter. 

Many  illustrations  are  found  in  nature,  such  as  the  bones  of 
animals,  the  quills  of  birds'  feathers,  the  straw  of  grain,  and  the 
tubular  stalks  of  some  larger  plants. 

An  interesting  application  of  the  principle  has  been  made  in 
modern  times  in  the  construction  of  iron  tubular  bridges. 


200.  MISCELLANEOUS   PROBLEMS  IN    MECHANICS. 

1.  Two  forces,  F  and  F',  acting  in  the  diagonals  of  a  parallelo- 
gram, keep  it  at  rest  in  such  a  position  that  one  of  its  edges  is 
horizontal ;  show  that  F  sec  a'  —  F'  sec  a  =  W  cosec  (a  +  a'), 
where  W  is  the  weight  of  the  parallelogram,  and  a  and  a'  the 
angles  between  the  diagonals  and  the  horizontal  side. 
9 


130  MECHANICS. 

2.  Four  parallel  forces  act  at  the  angles  of  a  plane  quadri- 
lateral, and  are  inversely  proportional  to  the  segments  of  its  diag- 
onals nearest  to  them ;  show  that  the  point  of  application  of  their 
resultant  lies  at  the  intersection  of  the  diagonals. 

3.  Find  the  centre  of  gravity  of  four  equal  heavy  particles 
placed  at  the  four  angular  points  of  a  triangular  pyramid. 

4.  Five  pieces  of  a  uniform  chain  are  hung  at  equidistant  points 
along  a  rigid  rod  without  weight,  and  their  lower  ends  are  in  a 
straight  line  passing  through  one  end  of  the  rod;  find  the  centre 
of  gravity  of  the  system. 

5.  A  right  square  pyramid,  whose  height  is  8  feot,  and  the 
edge  of  its  base  1  foot,  is  tipped  on  one  edge  till  it  is  on  the  point 
of  falling;  what  angle  does  its  axis  make  with  the  horizon  ? 

6.  If  three  uniform  rods  be  rigidly  united  so  as  to  form  half  of 
a  regular  hexagon,  prove  that  if  suspended  from  one  of  the  angles, 
one  of  the  rods  will  be  horizontal. 

7.  A  cone  of  uniform  density,  whose  slant  height  is  15  inches, 
is  suspended  by  the  edge  of  its  base,  when  its  axis  is  found  to  in- 
cline 12°  to  the  horizon ;  required  the  other  dimensions  of  the 
cone. 

8.  If  A  B  0  be  an  isosceles  triangle,  having  a  right  angle  at  C, 
and  if  D  and  E  be  the  middle  points  of  A  0  and  A  B,  respect- 
ively, prove  that  a  perpendicular  from  E  upon  B  D  will  pass 
through  the  centre  of  gravity  of  the  triangle  B  D  C. 

9.  An  oblique  cylinder,  inclining  62^°  to  the  horizon,  having 
slant  height  =  14  inches,  and  the  diameter  of  the  base  =  6A 
inches,  has  a  ball  of  the  same  material  hung  upon  its  edge,  which 
just  upsets  it;  required  the  diameter  of  the  ball. 

10.  A  body,  the  lower  surface  of  which  is  spherical,  rests  upon 
a  horizontal  plane ;  find  in  what  case  the  equilibrium  is  stable,  and 
in  what  case  unstable. 

11.  A  smooth  circular  ring  rests  on  two  pins  projecting  from  a 
wall,  and  the  pins  are  not  in  the  same  horizontal  plane ;  find  the 
pressure  on  each  pin. 

12.  A  given  isosceles  triangle  is  inscribed  in  a  circle ;  find  the 
centre  of  gravity  of  the  remaining  area  of  the  circle. 

13.  A  homogeneous  hemisphere  rests  with  its  convex  surface 
on  a  horizontal  plane ;  at  what  points  of  the  circumference  of  the 
plane  base  of  the  hemisphere  must  three  weights  of  10,  15,  and  20 
Ibs.  be  suspended,  in  order  that  its  position  be  not  changed  ? 

14.  Two  smooth  cylinders  of  equal  radii  just  fit  in  between  two 
parallel  vertical  walls,  and  rest  on  a  smooth  horizontal  plane,  with- 
out pressing  against  the  walls ;  if  a  third  equal  cylinder  be  placed 
on  the  top  of  them,  find  the  resulting  pressure  against  either 
wall. 


PROBLEMS    IN    MECHANICS.  131 

1 5.  A  cylinder,  suspended  by  a  point  on  the  side,  inclines  40° 
to  the  horizon ;  the  point  is  moved  3  feet  lengthwise  on  the  side, 
and  then  the  cylinder  inclines  24°  to  the  horizon,  with  the  other 
end  down ;  find  the  point  of  suspension,  that  the  cylinder  may 
hang  horizontal. 

16.  A  flat  semicircular  board,  with  its  plane  vertical,  and  curved 
edge  upward,  rests  on  a  smooth  horizontal  plane,  and  is  pressed  at 
two  given  points  of  its  circumference  by  two  heavy  rods,  which 
slide  freely  in  vertical  guides; 'find  the  ratio  of  the  weights  of  the 
rods,  that  the  board  may  be  in  equilibrium. 

17.  The  radii  of  a  wheel  and  axle  are  12  inches  and  3  inches; 
the  power  is  30  Ibs.,  the  weight  100  Ibs. :  as  the  power  in  this  case 
preponderates,  required  how  many  degrees  from  the  bottom  of  the 
wheel  the  end  of  the  rope  is  when  the  forces  are  in  equilibrium. 

18.  A  frustum  is  cut  from  a  right  cone  by  a  plane  bisecting  the 
axis,  and  parallel  to  the  base;  show  that  it  will  rest  with  its  slant 
side  on  a  horizontal  plane,  if  the  height  of  the  cone  have  to  the 
diameter  of  its  base  a  greater  ratio  than  Vl  to  Vvj. 

19.  Explain  the  action  of  an  oar,  when  used  in  rowing,  and 
determine  the  effect  produced,  having  given  the  distances  from 
the  hands  to  the  side  of  the  boat,  and  from  the  side  of  the  boat  to 
the  point  where  the  oar  may  be  considered  as  acting  on  the  water. 

20.  A  uniform  wheel,  free  to  revolve  on  its  axis,  has  the  weights, 
21  Ibs.  and  13  Ibs.,  attached  to  the  circumference,  100°  apart;  how 
far  from  the  bottom  will  the  weights  be,  respectively,  when  the 
system  is  in  equilibrium? 

21.  Two  equal  rods,  without  weight,  are  connected  at  their 
middle  points  by  a  pin,  which  allows  free  motion  in  a  vertical 
plane ;  they  stand  upon  a  horizontal  plane,  and  their  upper  ex- 
tremities are  connected  by  a  thread,  which   carries    a  weight. 
Show  that  the  weight  will  rest  half  way  between  the  pin  and  the 
horizontal  line  joining  the  upper  ends  of  the  rods. 

22.  A  uniform  heavy  rod,  of  given  length,  is  to  be  supported  in 
a  given  position,  with  its  upper  end  resting  against  a  smooth  ver- 
tical wall,  by  a  string  fastened  to  its  lower  end ;  find  the  point  in 
the  wall  to  which  the  string  must  be  attached. 

23.  A  light  cord,  with  one  end  attached  to  a  fixed  point,  passes 
over  a  pulley  in  the  same  horizontal  plane  with   the  fixed  point, 
and  supports  a  weight  hanging  freely  at  its  other  end.    A  heavy 
ring  being  put  upon  the  cord  in  different  places  between  the  fixed 
point  and  the  pulley,  it  is  required  to  show  that,  if  the  weight  of 
the  ring  be  small  compared  with  the  other  weight,  the  positions 
of  the  ring,  when  in  equilibrium,  will  be  approximately  in  the  &rc 
of  a  circle. 

24.  If  particles  of  unequal  weight  be  placed  in  the  angular 


132  MECHANICS. 

points  of  a  triangular  pyramid,  and  G  be  their  common  centre  of 
gravity,  G',  G",  G'",  &c.,  be  the  common  centres  of  gravity  for 
every  possible  arrangement  of  the  particles ;  show  that  the  centre 
of  gravity  of  equal  particles,  placed  at  G,  G',  G",  &c.,  is  the  centre 
of  gravity  of  the  pyramid. 

25.  Two  equal  circular  disks  with  smooth  edges,  placed  on 
their  flat  sides  in  the  corner  between  two  smooth  vertical  planes, 
inclined  to  each  other  at  a  given  angle,  touch  each  other  in  the 
line  bisecting  that  angle;  find  the. radius  of  the  least  disk  that 
may  be  pressed  between  them  without  causing  them  to  separate. 

26.  A  ladder  of  uniform  weight  throughout,   36  feet  long, 
weighs  72  Ibs.,  and  leans  against  a  vertical  wall,  making  an  angle 
of  66°  40'  with  the  horizon ;  a  man,  weighing  130  Ibs.,  ascends  30 
feet  on  the  ladder ;  required  the  amount  of  pressure  against  the 
wall. 

27.  Where  is  the  centre  of  gravity  of  the  area  included  between 
two  circles  tangent  to  each  other  internally  ? 


PART   II. 

HYDROS  TA.TICS. 


CHAPTER  I. 

LIQUIDS    AT    REST. 

201.  Liquids  Distinguished  from  Solids  and  Gases.— 

A.  fluid  is  a  substance  whose  particles  are  moved  among  each  other 
by  a  very  slight  force.  In  solid  bodies  the  particles  are  held  by 
the  force  of  cohesion  in  fixed  relations  to  each  other;  hence  such 
bodies  retain  their  form  in  spite  of  gravity  or  other  small  forces 
exerted  upon  them.  If  a  solid  be  reduced  to  the  finest  powder, 
still  each  grain  of  the  powder  is  a  solid  body,  and  its  atoms  are 
held  together  in  a  determinate  shape.  A  pulverized  solid,  if  piled 
up,  will  settle  by  the  force  of  gravity  to  a  certain  inclination,  ac- 
cording to  the  smallness  and  smoothness  of  its  particles,  while  a 
liquid  will  not  rest  till  its  surface  is  horizontal. 

Fluids  are  of  two  kinds,  liquids  and  gases.  In  a  liquid,  there 
is  a  perceptible  cohesion  among  its  particles ;  but  in  a  gas,  the 
particles  mutually  repel  each  other.  These  fluids  are  also  distin- 
guished by  the  fact  that  liquids  cannot  be  compressed  except  in  a 
very  slight  degree,  while  the  gases  are  very  compressible.  A  force 
of  15  pounds  on  a  square  inch,  applied  to  a  mass  of  water,  will 
compress  it  only  about  .000046  of  its  volume,  as  is  shown  by  an 
instrument  devised  by  Oersted.  But  the  same  force  applied  to  a 
quantity  of  air  of  the  usual  density  at  the  earth's  surface  will  re- 
duce it  to  one-half  of  its  former  volume. 

202.  Transmitted  Pressure.— It  is  an  observed  property  of 
fluids  that  a  force  which  is  applied  to  one 

part  is  transmitted  undivided  to  all  parts. 
For  instance,  if  a  piston  A  (Fig.  143)  is 
pressed  upon  the  water  in  the  vessel  AD G 
with  a  force  of  one  pound,  every  other  pis- 
ton of  the  same  size,  as  B,  O,  D,  or  E,  re- 
ceives a  pressure  of  one  pound  in  addition 
to  the  previous  pressure  of  the  water  itself. 
Hence  the  whole  amount  of  bursting  press- 
ure exerted  within  the  vessel  by  the  weight 


134 


HYDROSTATICS. 


upon  A  equals  as  many  pounds  as  there  arc  portions  of  surface 
equal  to  the  area  of  A.  And  if  the  pressure  is  increased  till  the 
vessel  bursts,  the  fracture  is  as  likely  to  occur  in  some  other  part 
as  in  that  toward  which  the  force  is  directed. 

203.  The  Hydraulic  Press. — An  important  application  of 
the  principle  of  transmitted  pressure  occurs  in  Bramah's  hydraulic 
press,  represented  in  Fig.  144.  The  walls  of  the  cylinder  and  res- 

FIG.  144. 


ervoir  are  partly  removed,  to  show  the  interior.  A  is  a  small 
forcing  pump,  worked  by  the  lever  M,  by  which  water  is  raised  in 
the  pipe  a  from  the  reservoir  H,  and  driven  through  the  tube  K 
into  the  cylinder  B,  where  it  presses  up  the  piston  P,  and  the  iron 
plate  on  the  top  of  it,  against  the  substance  above.  At  each  down- 
ward stroke  of  the  small  piston  p,  a  quantity  of  water  is  transferred 
to  the  cylinder  B,  and  presses  up  the  large  piston  with  a  force  as 
many  times  greater  than  that  exerted  on  the  small  one  as  the 
under  surface  of  P  is  greater  than  that  of  p  (Art.  202).  If.  the 
diameter  of  p  is  one  inch,  and  that  of  P  is  ten  inches,  then  any 
pressure  on^>  exerts  a  pressure  100  times  as  great  on  P.  The  le vei- 
l/gives an  additional  advantage.  If  the  distances  from  the  ful- 
crum to  the  rod  p  and  to  the  hand  are  as  1  :  5,  this  ratio  com- 
pounded with  the  other,  1  :  100,  gives  the  ratio  of  power  at  M  to 


EQUILIBRIUM    OF    A    FLUID.  135 

the  pressure  at  Q  as  1 :  500;  so  that  a  power  of  100  Ibs.  exerts  a 
pressure  of  50000  Ibs. 

This  machine  has  the  special  advantage  of  working  with  a 
small  amount  of  friction.  It  is  used  for  pressing  paper  and  books, 
packing  cotton,  hay,  &c. ;  also  for  testing  the  strength  of  cables 
and  steam-boilers.  It  has  been  sometimes  employed  to  raise  great 
weights,  as,  for  instance,  the  tubular  bridge  over  the  Menai  straits ; 
the  two  portions,  after  being  constructed  at  the  water  level,  were 
raised  more  than  100  feet  to  the  top  of  the  piers,  by  two  hydraulic 
presses.  The  weight  of  each  length  lifted  at  once  was  more  than 
1800  tons. 

The  relation  of  power  to  weight  in  the  hydraulic  press  is  in 
accordance  with  the  principle  of  virtual  velocities  (Art.  142).  For, 
while  a  given  quantity  of  water  is  transferred  from  the  smaller  to 
the  larger  cylinder,  the  velocity  of  the  large  piston  is  as  much  less 
than  that  of  the  small  one  as  its  area  is  greater.  But  we  have  seen 
that  the  pressures  are  directly  as  the  areas.  Therefore,  in  this  as 
in  other  machines,  the  intensities  of  the  forces  are  inversely  as 
their  virtual  velocities. 

204.  Equilibrium  of  a  Fluid.— In  order  that  a  fluid  may 
be  at  rest, 

1.  The  pressures  at  any  one  point  must  be  equal  in  all  direc- 
tions. 

2.  The  surface  must  be  perpendicular  to  the  resultant  of  the 
forces  which  act  upon  it. 

Both  of  these  conditions  result  from  the  mobility  of  the  par- 
ticles. It  is  obvious  that  the  first  must  be  true,  since,  if  any 
particle  were  pressed  more  in  one  dirf  ction  than  another,  it  would 
move  in  the  direction  of  the  greater  force,  and  therefore  not  be  at 
rest,  as  supposed. 

In  order  to  show  the  truth  of  the  second  condition,  let  mp 
(Fig.  145)  represent  the  resultant  of 
the  forces  which  act  on  the  fluid.  Then,  FIG.  145. 

if  the  surface  is  not  perpendicular  to 
m  p,  that  force  may  be  resolved  into 
m  q  perpendicular  to  the  surface,  and 
m  f  parallel  to  it.  The  latter,  mf,  not 
being  opposed,  the  particles  move  in 
that  direction. 

As  gravity  is  the  principal  force  which  acts  on  all  the  particles, 
the  surface  of  a  fluid  at  rest  is  ordinarily  level,  that  is,  perpendicu- 
lar to  a  vertical  or  plumb  line.  If  the  surface  is  of  small  extent, 
it  is  sensibly  a  plane,  though  it  is  really  curved,  because  the  verti- 
cal lines,  to  which  it  is  perpendicular,  converge  toward  the  centre 
of  the  earth. 


136 


HYDROSTATICS. 


205.  The  Curvature  of  a  Liquid  Surface.  —  The  earth 
being  7912  miles  in  diameter,  a  distance  of  100  feet  on  its  surface 
subtends  an  angle  of  about  one  second  at  the  centre,  and  therefore 
the  levels  of  two  places  100  feet  apart  are  inclined  one  second  to 
each  other. 

The  amount  of  depression  for  moderate  distances  is  found  by 
the  formula,  d  =  f  L*,  in  which  d  is  the  de- 
pression in  feet,  and  L  the  length  of  arc  in 
miles.  Let  B  E  (Fig.  146)  be  a  small  arc  of 
a  great  circle  on  the  earth  ;  then  C  E  is  the 
depression.  As  B  E  is  small,  its  chord  may 
be  considered  equal  to  the  arc,  and  B  G  equal 
to  the  depression.  But  B  G  :  B  E  :  :  BE: 

B  A  ;  that  is,  d  :  L  :  :  L  :  7912;  or  d  = 


In  order  to  express  d  in  feet,  while  the  other 
lines  are  in  miles,  we  have 


x  52805 


7912  x  5280 


L*  x  5280        0  ra  , 

—ij$l2 =  *  L>  verJ  nearll- 


This  gives,  for  one  mile,  d  =  8  inches ;  for  two  miles,  d  =  2 
ft.  8  in.;  and  for  100  miles,  d  =  6667  ft,  &c.  If  a  canal  is  100 
miles  long,  each  end  is  more  than  a  mile  below  the  tangent  to  the 
surface  of  the  water  at  the  other  end. 

206.  The  Spirit  Level.— Since  the  surface  of  a  liquid  at 
rest  is  level,  any  straight  line  which  is  placed  parallel  to  such  a 
surface  is  also  level.  Leveling  instruments  are  constructed  on 
this  principle.  The  most  accurate  kind  is  the  one  called  the 
spirit  level.  Its  most  essential  -pIG  ^^ 

part  is  a  glass  tube,  A  B  (Fig. 
147),  nearly  filled  with  alcohol 
(because  water  would  be  liable  to  freeze),  and  hermetically  sealed. 
The  tube  having  a  little  convexity  upward  from  end  to  end, 
though  so  slight  as  not  to  be  visible,  the  bubble  of  air  moves  to 
the  highest  part,  and  changes  its  place  by  the  least  inclination  of 
the  tube.  The  tube  is  so  connected  with  a  straight  bar  of  wood 
or  metal,  as  D  C  (Fig.  148),  or  for  nicer  purposes,  with  a  telescope, 
that  the  bubble  is  at  the  FlG  148 

middle  M  when  the  bar 
or  the  axis  of  the  tele- 
scope is  exactly  level. 
The  tube  usually  has 
graduation  lines  upon  it 
for  adjusting  the  bubble  accurately  to  the  middle. 


PRESSURE    AS    DEPTH. 


137 


207.  Pressure  as  Depth. — From  the  principle  of  equal 
transmission  of  force  in  a  fluid,  it  follows  that,  if  a  liquid  is  uni- 
formly dense,  its  pressure  on  a  given  area  varies  as  the  perpen- 
dicular depth,  whatever  the  form  or  size  of  the  reservoir.  Let  the 
vessel  A  B  C  D  (Fig.  149),  having  the  form  of  a  right  prism,  be 
filled  with  water,  and  imagine  the  water  to  be  divided  by  horizon- 
tal planes  into  strata  of  equal  thickness.  If  the  density  is  every- 
where the  same,  the  weights  of  these  strata  are  equal.  But  the 
pressure  on  each  stratum  is  the  sum  of  the  weights  of  all  tho 
strata  above  it.  Therefore,  in  this  case,  the  pressure  varies  as  the 
depth. 

FIG.  149.  FIG.  150  FIG.  151 

JL         C      V        3 


CJ) 


But  let  the  reservoir  A  B  E  H  (Fig.  150)  contain  water  which 
is  not  directly  beneath  the  highest  part.  The  pressures  in  the 
column  A  B  C  D  are  transmitted  laterally  to  E  H,  however  far 
distant ;  so  that  the  surface  of  each  horizontal  stratum  sustains 
equal  pressures  in  all  parts,  whether  directly  beneath  A  B  or  not. 
Hence,  if  G  H  is  equal  to  C  D,  the  downward  pressure  on  G  H  is 
equal  to  the  weight  of  A  B  C  D ;  so,  also,  the  upward  pressure  on 
E  F  is  equal  to  the  weight  of  A  B  L  M,  and  would  just  sustain 
the  column  of  water  E  F  N  P. 

Again,  if  the  base  is  smaller  than  the  top,  as  in  the  vessel 
A  BE  F  (Fig.  151),  then  the  pressure  on  E  F  equals  only  the 
weight  of  the  column  G  D  E  F.  The  water  in  the  surrounding 
space  ACE,  B  D  F,  simply  serves  as  a  vertical  wall  to  balance 
the  lateral  pressures  of  the  central  column. 

If  the  surface  pressed  upon  is  oblique  or  vertical,  then  the 
points  of  it  are  at  unequal  depths ;  in  this  case,  the  depth  of  the 
area  is  understood  to  be  the  average  depth  of  all  its  parts ;  that 
is,  the  depth  of  its  centre  of  gravity. 

If  the  fluid  were  compressible,  the  lower  strata  would  be  moro 
dense  than  the  upper  ones,  and  therefore  the  pressure  would  in- 
crease at  a  faster  rate  than  the  depth. 

208.  Amount  of  Pressure  in  Water.— One  cubic  foot  of 
water  weighs  1000  ounces,  or  62.5  pounds.  Therefore,  the  pressure 
on  one  square  foot,  at  the  depth  of  one  foot,  is  62.5  pounds.  From 
tli is,  as  the  unit  of  hydrostatic  pressure,  it  is  easy  to  determine  tho 


138 


HYDROSTATICS. 


n'cssurcs  on  all  surfaces,  at  all  depths ;  for  it  is  obvious  that,  when 
the  depth  is  the  same,  the  pressure  varies  as  the  surface  pressed 
upon ;  and  it  has  been  shown  that,  on  a  given  surface,  the  press- 
ure varies  as  the  depth  of  its  centre  of  gravity ;  it  therefore  varies 
as  the  product  of  the  two.  Let  p  —  pressure ;  a  =  area  pressed 
upon;  and  d  =  the  depth  of  its  centre  of  gravity;  then 
p  —  a  d  x  62.5. 

Depth.                        Pounds  per  sq.  ft. 
I  ft 62.5 


10 625 

16 IOOO 


Depth.  Pounds  per  sq.  ft. 

100  ft 6,250 

I  mile 330.000 

5  miles 1,650,000 


FIG.  152. 


From  the  above  table  it  may  be  inferred  that  the  pressure  on 
a  square  foot  in  the  deepest  parts  of  the  ocean  must  be  not  far 
from  two  millions  of  pounds;  for  the  depth  in  some  places  is 
more  than  five  miles,  and  sea-water  weighs  64.37  pounds,  instead 
of  62.5  pounds.  A  brass  vessel  full  of  air,  containing  only  a  pint, 
and  whose  walls  were  one  inch  thick,  has  been  known  to  be 
crushed  in  by  this  great  pressure,  when 
sunk  to  the  bottom  of  the  ocean. 

Owing  to  the  increase  of  pressure 
with  depth,  there  is  great  difficulty  in 
confining  a  high  column  of  water  by 
artificial  structures.  The  strength  of 
banks,  dams,  flood-gates,  and  aqueduct 
pipes,  must  increase  in  the  same  ratio 
as  the  perpendicular  depth  from  the  sur- 
face of  the  water,  without  regard  to  its 
horizontal  extent. 

209.  Column  of  Water  whose 
Weight  Equals  the  Pressure. — A 

convenient  mode  of  conceiving  readily 
of  the  amount  of  pressure  on  an  area, 
in  any  given  circumstances,  is  this: 
consider  the  area  pressed  upon  to  form 
the  horizontal  base  of  a  hollow  prism ; 
let  the  height  of  the  prism  equal  the 
average  depth  of  the  area;  and  then 
suppose  it  filled  with  water.  The  weight 
of  this  column  of  water  is  equal  to  the 
pressure.  For  the  contents  of  the  prism 
(whose  base  —  #,  and  its  height  =  d), 
=  a  d ;  and  the  weight  of  the  same  = 
a  d  x  62.5  Ibs. ;  which  is  the  same  ex- 
pression as  was  obtained  above  for  the 
pressure. 


LEVEL    IN    CONNECTED    VESSELS. 


139 


On  the  bottom  of  a  cubical  vessel  full  of  water,  the  pressure 
equals  the  weight  of  the  water;  on  each  side  of  the  same  the 
pressure  is  one-half  the  weight  of  the  water ;  hence,  on  all  the  five 
sides  the  pressure  is  three  times  the  weight  of  the  water ;  and  if 
the  top  were  closed,  on  which  the  pressure  is  zero,  the  pressure  on 
the  six  sides  is  the  same,  three  times  the  weight  of  the  water. 

210.  Illustrations   of  Hydrostatic  Pressure. — A  vessel 
may  be  formed  so  that  both  its  base  and  height  shall  be  great,  but 
its  cubical  contents  small ;  in  which  case,  a  great  pressure  is  pro- 
duced by  a  small  quantity  of  water.     The  hydrostatic  bellows  is 
an  example.    In  Fig.  152,  the  weight  which  can  be  sustained  on 
the  lid  D  I  by  the  column  A  D  is  equal  to  that  of  a  prism  or  cyl- 
inder of  water,  whose  base  is  D  /,  and  its  height  D  A.    It  is  im- 
material how  shallow  is  the  stratum  of  water  on  the  base,  or  how 
slender  the  tube  A  D,  if  greater  than  a  capillary  size. 

In  like  manner,  a  cask,  after  being  filled,  may  be  burst  by  an 
additional  pint  of  water ;  for,  by  screwing  a  long  and  slender  pipe 
into  the  top  of  the  cask,  and  filling  it  with  water,  the  pressure  is 
easily  made  greater  than  the  strength  of  the  cask  can  bear. 

211.  The  Same  Level  in  Connected  Vessels. — In  tubes 
or  reservoirs  which  communicate  with  each  other,  water  will  rest 
only  when  its  surface  is  at  the 

same  level  in  them  all.  If  water 
is  poured  into  D  (Fig.  153),  it 
will  rise  in  the  vertical  tube  B, 
so  as  to  stand  at  the  same  level 
as  in  D.  For,  the  pressure  to- 
ward the  right  on  any  cross-sec- 
tion E  of  the  horizontal  pipo 
m  n  equals  the  product  of  its 
area  by  its  depth  below  D.  So 
the  pressure  on  the  same  section 
towards  the  left  equals  the  pro- 
duct of  its  area  by  its  depth  be- 
low B.  But  these  pressures  are 
equal,  since  the  liquid  is  afc 
rest.  Therefore  E  is  at  equal  depths  below  B  and  Z);  in  other 
words,  B  and  D  are  on  the  same  level.  The  same  reasoning  ap- 
plies to  the  irregular  tubes  A  and  C,  and  to  any  others,  of  what- 
ever form  or  size. 

Water  conveyed  in  aqueducts,  or  running  in  natural  channels 
in  the  earth,  will  rise  just  as  high  as  the  source,  but  no  higher. 

Artesian  wells  illustrate  the  same  tendency  of  water  to  rise  to 
its  level  in  the  different  branches  of  a  tube.     When  a  deep  boring 


FIG  153. 


140  HYDROSTATICS. 

is  made  in  the  earth,  it  may  strike  a  layer  or  channel  of  water 
which  descends  from  elevated  land,  sometimes  very  distant.  The 
pressure  causes  it  to  rise  in  the  tube,  and  often  throws  it  many 
feet  above  the  surface.  Fig.  154  shows  an  artesian  well,  through 
which  is  discharged  the  water  that  descends  in  the  porous  stratum 
K  K,  confined  between  the  strata  of  clay  A  B  and  G  D. 

FIG.  154. 


212.  Centre  of  Pressure. — The  centre  of  pressure  of  any 
surface  immersed  in  water  is  that  point  through  which  passes  the 
resultant  of  all  the  pressures   on  the  surface.    It  is  the  point, 
therefore,  at  which  a  single  force  must  be  applied  in  order  to 
counterbalance  all  the  pressures  exerted  on  the  surface.    If  the 
surface  be  a  plane,  and  horizontal,  the  centre  of  pressure  coincides 
with  the  centre  of  gravity,  because  the  pressures  are  equal  on  every 
part  of  it,  just  as  the  force  of  gravity  is.     But  if  the  plane  surface 
makes  an  angle  with  the  horizon,  the  centre  of  pressure  is  lower 
than  the  centre  of  gravity,  since  the  pressure  increases  with  the 
depth.    For  example,  if  the  vertical  side  of  a  vessel  full  of  water 
is  rectangular,  the  centre  is  one-third  of  the  distance  from  the 
middle  of  the  base  to  the  middle  of  the  upper  side.     If  triangular, 
with  its  base  horizontal,  the  centre  of  pressure  is  one-fourth  of  the 
distance  from  the  middle  of  the  base  to  the  vertex.     If  triangular, 
with  the  top  horizontal,  the  centre  of  pressure  is  half  way  up  on 
the  bisecting  line. 

[See  Appendix  for  calculations  of  the  place  of  the  centre  of 
pressure.] 

213.  The  Loss  of  Weight   in  Water. — When  a  body  is 
immersed  in  water,  it  suffers  a  pressure  on  every  side,  which  is 
proportional  to  the  depth.     Opposite  components  of  lateral  press- 
vires,  being  exerted  on  surfaces  at  the  same  depth,  balance  each 
other ;  but  this  cannot  be  true  of  the  vertical  pressures,  since  the 
top  and  bottom  of  the  body  are  at  unequal  depths.     The  upward 
pressure  on  the  bottom  exceeds  the  downward  pressure  on  the  top ; 


EQUILIBRIUM    OF    FLOATING    BODIES.  141 

and  this  excess  constitutes  the  buoyant  power  of  a  fluid,  which 
causes  a  loss  of  weight. 

A  body  immersed  in  ivater  loses  weight  equal  to  the  weight  of 
water  displaced. 

For  before  the  body  was  immersed,  the  water  occupying  the 
same  space  was  exactly  supported,  being  pressed  upward  more 
than  downward  by  a  force  equal  to  its  own  weight.  The  weight 
of  the  lody,  therefore,  is  diminished  by  this  same  difference  of 
pressures,  that  is,  by  the  weight  of  the  displaced  water. 

On  the  supposition  of  the  complete  incompressibility  of  water, 
this  loss  is  the  same  at  all  depths,  because  the  weight  of  displaced 
water  is  the  same.  As  water,  however,  is  slightly  compressible,  its 
buoyant  power  must  increase  a  little  at  great  depths.  Calling  the 
compression  .000046  for  one  atmosphere  (=34  feet  of  water),  the 
bulk  of  water  at  the  depth  of  a  mile  is  reduced  by  about  7|  -$,  and 
its  specific  gravity  increased  in  the  same  ratio ;  so  that,  possibly,  a 
body  might  sink  near  the  surface,  and  float  at  great  depths  in  the 
ocean.  But  this  is  not  probable  in  any  case,  since  the  same  com- 
pressing force  may  reduce  the  volume  of  the  solid  as  much  as  that 
of  the  water.  And,  furthermore,  the  increase  of  density  by  in- 
creased depth  is  so  slow,  that  even  if  solids  were  incompressible, 
most  of  those  which  sink  at  all  would  not  find  their  floating  placo 
within  the  greatest  depths  of  the  ocean.  For  example,  a  stone 
twice  as  heavy  as  water  must  sink  100  miles  before  it  could  float. 

214.  Equilibrium  of  Floating  Bodies. — If  the  body  which 
is  immersed  has  the  same  density  as  water,  it  simply  loses  its 
whole  weight,  and  remains  wherever  it  is  placed.  But  if  it  is  less 
dense  than  water,  the  excess  of  upward  pressure  is  more  than  suf- 
ficient to  support  it;  it  is,  therefore,  raised  to  the  surface,  and 
comes  to  a  state  of  equilibrium  after  partly  emerging.  In  order 
that  a  floating  body  may  have  a  stable  equilibrium,  the  three  fol- 
lowing conditions  must  be  fulfilled : 

1.  It  displaces  an  amount  of  water  whose  weight  is  equal  to  its 
own. 

2.  The  centre  of  gravity  of  the  lody  is  in  the  same  vertical  line 
with  that  of  the  displaced  water. 

3.  The  metacenter  is  higher  than  the  centre  of  gravity  of  the 
lody. 

The  reason  for  ihe  first  condition  is  obvious;  for  both  the  body 
and  the  water  displaced  by  it  are  sustained  by  the  same  upward 
pressures,  and  therefore  must  be  of  equal  weight. 

That  the  second  is  true,  is  proved  as  follows :  Let  C  (Fig.  155, 1 ) 
be  the  centre  of  gravity  of  the  displaced  water,  while  that  of  the 
body  is  at  G.  Now  the  fluid,  previous  to  its  removal,  was  sue- 


142 


HYDROSTATICS. 


tained  by  an  upward  force  equal  to  its  own  weight,  acting  through 
its  centre  of  gravity  (?;  and  the  same  upward  force  now  acts  upon 


FIG.  155. 


the  floating  body  through  the  same  point.  But  the  body  is  urged 
downward  by  gravity  in  the  direction  of  the  vertical  line  AGE. 
Were  these  two  forces  exactly  opposite  and  equal,  they  would  keep 
the  body  at  rest;  but  this  is  the  case  only  when  the  points  C  and 
G  are  in  the  same  vertical  line :  in  every  other  position  of  these 
points,  the  two  parallel  forces  tend  to  turn  the  body  round  on  a 
point  between  them. 

215.  The  Metacenter. — To  understand  the  third  condition, 
the  metaceiitcr  must  be  defined.  When  a  floating  body  is  slightly 
inclined  from  its  state  of  equilibrium,  as  in  Fig.  155,  2  and  3,  and 
a  vertical  is  drawn  through  the  new  centre  of  gravity  C  of  the  dis- 
placed water,  this  vertical  must  intersect  the  former  vertical  A  B ; 
the  intersection,  M,  is  called  the  metacenter.  When  the  centre  of 
gravity  of  the  body  G  is  lower  than  the  metacenter,  as  in  Fig.  155, 2, 
the  parallel  forces,  downward  through  G  and  upward  through  6", 
revolve  the  body  back  to  its  position  of  equilibrium,  which  is  then 
called  a  stable  equilibrium.  But  if  the  centre  of  gravity  of  the 
body  is  higher  than  the  metacenter,  as  in  Fig.  155,  3,  the  rotation 
is  in  the  opposite  direction,  and  the  body  is  upset,  the  equilibrium 
being  unstable.  Once  more,  if  the  centre  of  gravity  of  the  body  is 
at  the  metacenter,  the  body  rests  indifferently  in  any  position,  as, 
for  example,  a  sphere  of  uniform  density.  The  equilibrium  in  this 
case  is  called  neutral. 

If  only  the  first  condition  is  fulfilled,  there  is  no  equilibrium ; 
if  only  the  first  and  second,  the  equilibrium  is  unstable ;  if  all  the 
three,  the  equilibrium  is  stable. 

In  accordance  with  the  third  condition,  it  is  necessary  to  place 
the  heaviest  parts  of  a  ship's  cargo  in  the  bottom  of  the  vessel,  and 
sometimes,  if  the  cargo  consists  of  light  materials,  to  fill  the  bot- 
tom with  stone  or  iron,  called  ballast,  lest  the  masts  and  rigging 
should  raise  the  centre  of  gravity  too  high  for  stability.  On  the 
same  principle,  those  articles  which  are  prepared  for-life-preservers, 


SPECIFIC    GRAVITY.  143 

in  case  of  shipwreck,  should  be  attached  to  the  upper  part  of  the 
body,  that  the  head  may  be  kept  above  water.  The  danger  arising 
from  several  persons  standing  up  in  a  small  boat  is  quite  apparent ; 
for  the  centre  of  gravity  is  elevated,  and  liable  to  become  higher 
than  the  metacenter,  thus  producing  an  unstable  equilibrium. 

216.  Floating  in  a  Small  Quantity  of  "Water, — As  press- 
ure on  a  given  surface  depends  solely  on  the  depth,  and  not  at  all 
on  the  extent  or  quantity  of  water,  it  follows  that  a  body  will  float 
as  freely  in  a  space  slightly  larger  than  itself  as  on  the  open  water 
of  a  lake.    For  instance,  a  ship  may  be  floated  by  a  few  hogsheads 
of  water  in  a  dock  whose  form  is  adapted  to  it.     In  such  a  case,  it 
cannot  be  literally  true  that  the  displaced  water  weighs  as  much 
as  the  vessel,  when  all  the  water  in  the  dock  may  not  weigh  a 
hundredth  part  as  much.   The  expression  "  displaced  water  "  means 
the  amount  which  would  fill  the  place  occupied  by  the  immersed 
portion  of  the  body.    An  experiment  illustrative  of  the  above  is,  to 
float  a  tumbler  within  another  by  means  of  a  spoonful  of  water 
between. 

217.  Floating   of   Heavy   Substances.— A  body  of  the 
most  dense  material  may  float,  if  it  has  such  a  form  given  it  as  to 
exclude  the  water  from  the  upper  side,  till  the  required  amount  is 
displaced.     Ships  are  built  of  iron,  and  laden  with  substances  of 
greater  specific  gravity  than  water,  and  yet  ride  safely  on  the  ocean. 
A  block  of  any  heavy  material,  as  lead,  may  be  sustained  by  the 
upward  pressure  beneath  it,  provided  the  water  is  excluded  from 
the  upper  side  by  a  tube  fitted  to  it  by  a  water-tight  joint. 

218.  Specific  Gravity. — The  weight  of  a  body  compared 
with  the  weight  of  the  same  volume  of  the  standard,  is  called  its 
specific  gravity. 

Distilled  water,  at  about  39°  F.,  the  temperature  of  its  greatest 
density,  is  the  standard  for  all  solids  and  liquids,  and  common  air, 
at  32°,  for  gases.  Therefore  the  specific  gravity  of  a  solid  or  a 
liquid  body,  is  the  ratio  of  its  weight  to  the  weight  of  an  equal 
volume  of  water ;  and  the  specific  gravity  of  an  aeriform  body  is 
the  ratio  of  its  weight  to  the  weight  of  an  equal  volume  of  air. 
Hence,  to  find  the  specific  gravity  of  a  solid  or  liquid,  divide  its 
weight  by  the  weight  of  the  same  volume  of  water ;  but  in  the  case 
of  a  gas,  divide  by  the  weight  of  the  same  volume  of  air. 

219.  Methods  of  Finding  Specific  Gravity.— 

1.  For  a  solid  heavie'r  than  water,  divide  its  weight  ly  its  low 
of  weight  in  water. 

The  reason  for  this  rule  is  obvious.  The  weight  which  a  sub- 
merged body  loses  (Art.  213)  is  equal  to  the  weight  of  the  dis- 


144 


HYDROSTATICS. 


placed  water,  which  has,  of  course,  the  same  volume  as  the  body ; 
therefore,  dividing  by  the  loss  is  the  same  as  dividing  by  the  height 
of  the  same  volume  of  water. 

2.  For  a  solid  lighter  than  water,  divide  its  weight  by  its  iveight 
added  to  the  loss  it  occasions  to  a  heavier  body  previously  balanced 
in  water. 

For,  if  the  light  body  be  attached  to  a  body  heavy  enough  to 
sink  it,  it  loses  all  its  own  weight,  and  causes  loss  to  the  other 
which  was  previously  balanced.  And  the  whole  loss  equals  the 
weight  of  water  displaced  by  the  light  body.  Hence,  as  before,  we 
in  fact  divide  the  weight  of  the  body  by  the  weight  of  the  same 
volume  of  water. 

3.  For  a  liquid,  find  the  loss  which  a  body  sustains  weighed  in 
the  liquid  and  then  in  water,  and  divide  the  first  loss  by  the  second. 

For  the  first  loss  equals  the  weight  of  the  displaced  liquid,  and 
the  second  that  of  the  displaced  water;  and  the  volume  in  each 
case  is  the  same,  namely,  that  of  the  body  weighed  in  them. 

But  the  specific  gravity  of  a  liquid  may  be  more  directly  ob- 
tained by  measuring  equal  volumes  of  it  and  of  water  in  a  flask, 
and  finding  the  weight  of  each.  Then  the  weight  of  the  liquid 
divided  by  that  of  the  water  is  the  specific  gravity  required. 

220.  The  Hydrometer,  or  Areometer. — In  commerce  and 
the  arts,  the  specific  gravities  of  substances  are  obtained  in  a  more 
direct  and  sufficiently  accurate  way,  by  instruments  constructed 
for  the  purpose.  The  general  name  for  such  instruments  is  the 
hydrometer,  or  areometer.  But  other  names  are  given  to  such  as 
are  limited  to  particular  uses ;  as,  for  example,  the  alcoometer  for 
alcohol,  and  the  lactometer  for  milk.  The  hydrometer,  represented 
in  Fig.  156,  consists  of  a  hollow  ball,  with  a 
graduated  stem.  Below  the  ball  is  a  bulb  con- 
taining mercury,  which  gives  the  instrument  a 
stable  equilibrium  when  in  an  upright  position. 
Since  it  will  descend  until  it  has  displaced  a 
quantity  of  the  fluid  equal  in  weight  to  itself,  it 
will  of  course  sink  to  a  greater  depth  if  the  fluid 
is  lighter.  From  the  depths  to  which  it  sinks, 
therefore,  as  indicated  by  the  graduated  stem, 
the  corresponding  specific  gravities  are  esti- 
mated. 

Nicholson's  hydrometer  (Fig.  157)  is  the  most 
useful  of  this  class  of  instruments,  since  it  may 
be  applied  to  finding  the  specific  gravities  of 
solid  as  well  as  liquid  bodies.  In  addition  to 
the  hollow  ball  of  the  common  hydromoter,  it  is  furnished  at  the 


THE    HYDROMETER. 


145 


FIG.  157. 


top  with  a  pan  A  for  receiving  weights,  and  a  cavity  beneath  for 
holding  the  substance  under  trial.  The  instrument  is  so  adjusted 
that  when  1000  grains  are  placed  in  the  pan,  the  instrument  sinks 
in  distilled  water  at  the  temperature  of 
39^-°  F.  to  a  fixed  mark,  0,  on  the  stem. 
Calling  the  weight  of  the  instrument  W, 
the  weight  of  displaced  water  is  W  4-  1000. 
To  find  the  specific  gravity  of  a  liquid, 
place  in  the  pan  such  a  weight  w  as  will 
just  bring  the  mark  to  the  surface.  Then 
the  weight  of  the  liquid  displaced  is  W+  w. 
But  its  volume  is  equal  to  that  of  the  dis- 
placed water.  Therefore  its  specific  grav- 
W+w 


To  find  the  specific  gravity  of  a  solid, 
place  in  the  pan  a  fragment  of  it  weighing 
less  than  1000  grains,  and  add  the  weight 
w  required  to  sink  the  mark  to  the  water- 
level.  Then  the  weight  of  the  substance 
in  air  is  1000  —  w.  Kemove  the  substance 
to  the  cavity  at  the  bottom  of  the  instrument,  and  add  to  the 
weight  in  the  pan  a  sufficient  number  of  grains  w'  to  sink  the 
mark  to  the  surface.  Then  w'  is  the  loss  of  weight  in  water ; 

1000  —  w  .    ., 
therefore, -f is  the  specific  gravity  of  the  substance. 

221.  Specific  Gravity  of  Liquids  by  Means  of 
Heights. — The  specific  gravity  of  two  liquids  may  be  compared 
by  their  relative  heights  when  in  equilibrium.  Let  the  tubes  m 
and  n  (Fig.  158)  communicate  with 
each  other,  and  be  furnished  with 
a  scale  of  heights  above  the  zero 
line  B  C.  Suppose  the  column  of 
water  A  B  to  be  in  equilibrium 
with  the  column  of  mercury  CD. 
Put  h  =  the  height  of  the  water, 
h'  =  the  height  of  the  mercury,  and 
s  =  the  specific  gravity  of  the  lat- 
ter; then,  since  pressure  varies  as 
the  product  of  height  and  density, 
and  the  pressures  in  this  case  are 
equal,  we  have  Ti  x  1  =  h'  x  s ; 

whence  s  =  TT  ;  that  is,  the  specific 
10 


FIG.  158. 


146  HYDROSTATICS. 

gravity  is  found  ly  dividing  the  height  of  the  water  ty  the  height 
of  the  liquid.  Also,  h  :  li' : :  s  :  1;  that  is,  the  heights  of  two 
columns  in  equilibrium  are  inversely  as  their  specific  gravities. 

The  heavier  liquid  should  be  poured  in  first,  till  it  stands 
somewhat  above  B  C,  the  zero  mark  of  the  scale ;  and  then  the 
lighter  should  be  poured  into  one  branch,  till  it  presses  the  other 
down  to  the  zero  line.  The  heights  of  both  are  reckoned  upward 
from  B  (7,  since  the  heavy  liquid  below  B  C  balances  itself. 

222.  Table  of  Specific  Gravities. — An  accurate  knowl- 
edge of  the  specific  gravities  of  bodies  is  important  for  many  pur- 
poses of  science  and  art,  and  they  have  therefore  been  determined 
with  the  greatest  possible  precision.  The  heaviest  of  all  known 
substances  is  platinum,  whose  specific  gravity,  when  compressed 
by  rolling,  is  22,  water  being  1 ;  and  the  lightest  is  hydrogen,  whose 
specific  gravity  is  =  .073,  common  air  being  1.  Now,  as  water  is 
about  800  times  as  heavy  as  air,  it  is  (800  -r  .073  =)  10,959  times 
as  heavy  as  hydrogen.  Therefore  platinum  is  about  (10,959  x  22  — ) 
241,000  times  as  heavy  as  hydrogen.  Between  these  limits,  1  and 
241,000,  there  is  a  wide  range  for  the  specific  gravities  of  all  other 
substances.  As  a  class,  the  common  metals  are  the  heaviest 
bodies ;  next  to  these  come  the  metallic  ores ;  then  the  precious 
gems ;  minerals  in  general,  animal  and  vegetable  substances,  as 
shown  in  the  following  table; 

Metals  (pure),  not  including  the  bases  of  the  alkalies  and 

earths,  from 5 — 22 

Platinum     ....  22.0      Copper 8.90 


Gold 19.25 

Mercury 

Lead 11.35 

Silver 10.47 


Steel 7.84 

Iron 7.78 

Tin ;     .  7.29 

Zinc 7.00 


Metallic  ores,  lighter  than  the   pure  metals,  but  usually 

above 4.00 

Precious  gems,  as  the  ruby,  sapphire,  and  diamond    .     .     .     3 — 4 

Minerals,  comprehending  most  stony  bodies 2 — 3 

Liquids,  from  ether  highly  rectified  to  sulphuric  acid  highly 

concentrated f — 2 

Acids  in  general,  heavier  than  water. 
Oils  in  general,  lighter ;  but  the  oils  of  cloves  and  cinna- 
mon are  heavier  than  water  ;  the  greater  part  lie  between 

.9  and  i 9 — I 

Milk 1.032 

Alcohol  (perfectly  pure) 797 

"        of  commerce 835 

Proof  spirit 923 

Wines  ;  the  specific  gravity  of  the  lighter  wines,  as  Cham- 
pagne and  Burgundy,  is  a  little  less,  and  of  the  heavier 
wines,  as  Malaga,  a  little  greater  than  that  of  water. 
Woods,  cork  being  the  lightest,  and  lignum  vitze  the  heaviest  .24 — 1.34 


COHESION    AND    ADHESION.  147 

223.  Floating. — The  human  body,  when  the  lungs  are  filled 
with  air,  is  lighter  than  water,  and  but  for  the  difficulty  of  keeping 
the  lungs  constantly  inflated,  it  would  naturally  float.  "With  a  mod- 
erate degree  of  skill,  therefore,  swimming  becpmes  a  very  easy  pro- 
cess, especially  in  salt  water.    When,  however,  a  man  plunges,  as 
divers  sometimes  do,  to  a  great  depth,  the  air  in  the  lungs  becomes 
compressed,  and  the  body  does  not  rise  except  by  muscular  effort. 
The  bodies  of  drowned  persons  rise  and  float  after  a  few  days,  in 
consequence  of  the  inflation  occasioned  by  putrefaction. 

As  rocks  are  generally  not  much  more  than  twice  as  heavy  as 
water,  nearly  half  their  weight  is  sustained  while  they  are  under 
water ;  hence,  their  weight  seems  to  be  greatly  increased  as  soon 
as  they  are  raised  above  the  surface.  It  is  in  part  owing  to  their 
diminished  weight  that  large  masses  of  rock  are  transported  with 
great  facility  by  a  torrent.  While  bathing,  a  person's  limbs  feel  as 
if  they  had  nearly  lost  their  weight,  and  when  he  leaves  the  water, 
they  seem  unusually  heavy. 

224.  To  Find  the  Magnitude  of  an  Irregular  Body. — 

It  would  be  a  long  and  difficult  operation  to  find  the  exact  con- 
tents of  an  irregular  mineral  by  direct  measurement.  But  it 
might  be  found  with  facility  and  accuracy  by  weighing  it  in  air 
and  then  finding  its  loss  of  weight  in  water.  The  loss  is  the  weight 
of  a  mass  of  water  having  the  same  volume.  Now,  as  1000  ounces 
of  water  measure  1728  cubic  inches,  a  direct  proportion  will  show 
what  is  the  volume  of  the  displaced  water ;  that  is,  of  the  mineral 
itself. 

225.  Cohesion    and    Adhesion. — What    distinguishes    a 
liquid  from  a  solid  is  not  its  want  of  cohesion  so  much  as  the 
mobility  of  its  particles.     It  is  proved  in  many  ways  that  the  par- 
ticles of  a  liquid  strongly  attract  each  other.    It  is  owing  to  this 
that  water  so  readily  forms  itself  into  drops.    The  same  property 
is  still  more  observable  in  mercury,  which,  when  minutely  divided, 
will  roll  over  surfaces  in  spherical  forms.    When  a  disk  of  almost 
any  substance  is  laid  upon  water,  and  then  raised  gently,  it  lifts  a 
column  of  water  after  it  by  adhesion,  till  at  length  the  edge  of  the 
fluid  begins  to  .divide,  and  the  column  is  detached,  not  in  all  parts 
at  once,  but  by  a  successive  rupturing  of  the  lateral  surface.     It  is 
proved  that  the  whole  attraction  of  the  liquid  would  be  far  too 
great  to  be  overcome  by  the  force  applied  to  pull  off  the  disk,  were 
it  not  that  it  is  encountered  by  little  and  little,  at  the  edges  of  the 
column.    But  it  is  the  cohesion  of  the  water  which  is  overcome  in 
this  experiment ;  for  the  upper  lamina  still  adheres  to  the  disk. 
By  a  pair  of  scales  we  find  that  it  requires  the  same  force  to  draw 
off  disks  of  a  given  size,  whatever  the  materials  may  be,  provided 


148 


HYDROSTATICS. 


they  are  wet  when  detached.  This  is  what  might  be  expected, 
since  in  each  case  we  break  the  attraction  between  two  laminae  of 
water.  Bnt  if  we  use  disks  which  are  not  wet  by  the  liquid,  it  is 
not  generally  true  that  those  of  different  material  will  be  removed 
by  the  same  force ;  indicating  that  some  substances  adhere  to  a 
given  liquid  more  strongly  than  others. 

These  molecular  attractions  extend  to  an  exceedingly  small 
distance,  as  is  proved  by  many  facts.  A  lamina  of  water  adheres 
as  strongly  to  the  thinnest  disk  that  can  be  used  as  to  a  thick  one ; 
BO,  also,  the  upper  lamina  coheres  with  equal  force  to  the  next 
below  it,  whether  the  layer  be  deep  or  shallow. 

226.  Capillary  Action. — This  name  is  given  to  the  molecular 
forces,  adhesion  and  cohesion,  when  they  produce  disturbing 
effects  on  the  surface  of  a  liquid,  elevating  it  above  or  depressing 
it  below  the  general  level.  These  effects  are  called  capillary,  be- 
cause most  strikingly  exhibited  in  very  fine  (hair-sized)  tubes. 

The  liquid  will  be  elevated  in  a  concave  curve,  or  depressed  in  a 
convex  curve,  by  the  side  of  the  solid,  according  as  the  attraction  of 
the  liquid  molecules  for  each  other  is  less  or  greater  than  twice  the 
attraction  between  the  liquid  and  the  solid. 

Case  1st.  Let  H  K  (Fig.  159,  1)  and  L  M  be  a  section  of  the 
vertical  side  of  a  solid,  and  of  the  general  level  of  the  liquid.  The 


=Jf  Z 


particle  A,  where  these  lines  meet,  is  attracted  (so  far  as  this  sec- 
tion is  concerned)  by  all  the  particles  of  an  insensibly  small  quad- 
rant of  the  liquid,  the  resultant  of  which  attractions  is  in  the  line 
A  D,  45°  below  A  M.  It  is  also  attracted  by  all  the  particles  in 
two  quadrants  of  the  solid,  and  the  resultants  are  in  the  directions 
A  B,  45°  above,  and  A  B',  45°  below  L  M. 

Now  suppose  the  force  A  D  to  be  less  than  twice  A  B  or  A  Br. 
Cut  off  C  D  =  A  B;  then  A  B,  being  opposite  and  equal  to  CD, 
is  in  equilibrium  with  it.  The  remainder  A  C,  being  less  than 
A  B',  their  resultant  A  E  will  be  directed  toward  the  solid ;  and 
therefore  the  surface  of  the  liquid,  since  it  must  be  perpendicular 


CAPILLARY    TUBES    AND    PLATES.  149 

to  the  resultant  of  forces  acting  on  it  (Art.  204),  takes  the  direc- 
tion represented ;.  that  is,  concave  upward. 

Case  3d.  Let  A  D  (Fig.  159,  2),  the  attraction  of  A  toward 
the  liquid  particles,  be  more  than  twice  A  B,  the  attraction  toward 
a  quadrant  of  the  solid.  Then,  making  G  D  equal  to  A  B,  these 
two  resultants  balance  as  before ;  and  as  A  C  is  greater  than  A  B', 
the  angle  between  A  C  and  the  resultant  A  E  is  less  than  45°,  and 
A  is  drawn  away  from  the  solid.  Therefore  the  surface,  being 
perpendicular  to  the  resultant  of  the  molecular  forces  acting  on  it, 
is  convex  upward. 

Case  3d.  If  A  D  (Fig.  159,  3)  be  exactly  twice  A  B,  then  CD 
balances  A  B,  and  the  resultant  of  A  C  and  A  B'  is  A  E  in  a  ver- 
tical direction ;  therefore  the  surface  at  A  is  level,  being  neither 
elevated  nor  depressed. 

Case  1st  occurs  whenever  a  liquid  readily  wets  a  solid,  if 
brought  in  contact  with  it,  as,  for  example,  water  and  clean  glass. 
Case  2d  occurs  when  a  solid  cannot  be  wet  by  a  liquid,  as  glass  and 
mercury.  Case  3d  is  rare,  and  occurs  at  the  limit  between  the 
other  two ;  water  and  steel  afford  as  good  an  example  as  any. 

227.  Capillary  Tubes. — In  fine  tubes  these  molecular  forces 
affect  the  entire  columns  as  well  as  their  edges.     If  the  material 
of  the  tube  can  be  wet  by  a  liquid,  it  will  raise  a  column  of  that 
liquid  above  the  level,  at  the  same  time  making  the  top  of  the 
column  concave.    If  it  is  not  capable  of  being  wet,  the  liquid  is 
depressed,  and  the  top  of  the  column  is  convex.    The  first  case  is 
illustrated  by  glass  and  water ;  the  second  by  glass  and  mercury. 

The  materials  being  given,  the  distance  by  which  the  liquid  is 
elevated  or  depressed  varies  inversely  as  the  diameter.  Therefore 
the  product  of  the  two  is  constant. 

The  amount  of  elevation  and  depression  depends  on  the 
strength  of  the  molecular  forces,  rather  than  on  the  specific 
gravity  of  the  liquids.  Alcohol,  though  lighter  than  water,  is 
raised  only  half  as  high  in  a  glass  tube. 

If  the  upper  part  of  a  tube  is  capillary,  while  the  lower  part  is 
large,  a  liquid  is  sustained  (after  being  raised  by  suction)  at  the 
same  height  as  if  the  whole  were  capillary.  But  it  is  found  that 
the  large  mass  in  the  lower  part  is  upheld  by  atmospheric  pressure 
after  the  capillary  part  has  been  closed  by  the  molecular  attraction. 

228.  Parallel    and    Inclined    Plates.— Between  parallel 
plates  a  liquid  rises  or  falls  half  as  far  as  in  a  tube  of  the  same 
diameter.    This  is  because  the  sustaining  force  acts  only  on  two 
sides  of  each  filament,  while  in  a  tube  it  acts  on  all  sides.    There- 
fore, as  in  tubes  the  height  varies  inversely  as  the  diameter,  so  in 
plates  the  height  varies  inversely  as  the  distance  between  them. 


150 


HYDROSTATICS. 


FIG. 


If  the  plates  are  inclined  to  each  other,  having'  their  edge  of 
meeting  perpendicular  to  the  horizon,  the  surface  of  a  liquid  rising 
between  them  assumes  the  form  of  a  hyperbola,  whose  branches 
approach  the  vertical  edge,  and  the  water-level,  as  the  asymptotes 
of  the  curve.  This  results  from  the  law  already  stated,  that  the 
height  varies  inversely  as  the  distance  between  the  plates.  Let 
the  edge  of  meeting,  A  #(Fig.  160), 
be  the  axis  of  ordinates,  and  the 
line  in  which  the  level  surface  of 
the  water  intersects  the  glass,  A  P, 
the  axis  of  abscissas.  Let  B  C, 
D  E,  be  any  ordinates,  and  A  B, 
A  D,  their  abscissas,  and  B  L,  D  K, 
the  distances  between  the  plates. 
By  the  law  of  capillarity,  the  heights 
B  0,  D  E,  are  inversely  as  B  L,  D  K. 
But,  by  the  similar  triangles,  A  B  L, 


therefore,  B  C,  D  E,  are  inversely  as  A  B,  A  D  ;  and  this  is  a 
property  of  the  hyperbola  with  reference  to  the  centre  and  asymp- 
totes, that  the  ordinates  are  inversely  as  the  abscissas. 

229.  Effects  of  Capillarity  on  Floating  Bodies.—  Some 
cases  of  apparent  attractions  and  repulsions  between  floating 
bodies  are  caused  by  the  forms  which  the  liquid  assumes  on  the 
sides  of  the  bodies.  If  two  balls  raise  the  water  about  them,  and 
are  so  near  to  each  other  that  the  concave  surfaces  between  them 
meet  in  one,  they  immediately  approach  each  other  till  they  touch; 
and  then,  if  either  be  moved,  the  other  will  follow  it.  The  water, 
which  is  raised  and  hangs  suspended  between  them,  draws  them 
together. 

Again,  if  each  ball  depresses  the  water  around  it,  they  will  also 
move  to  each  other,  and  be  held  together,  so  soon  as  they  are  near 
enough  for  the  convex  surfaces  to  meet.  In  this  case,  they  are  not 
pulled,  but  pushed  together  by  the  hydrostatic  pressure  of  the 
higher  water  on  the  outside. 

Once  more,  if  one  ball  raises  the  water,  and  the  other  depresses 
it,  and  they  are  brought  so  near  each  other  that  the  curves  meet, 
they  immediately  move  apart,  as  if  repelled.  For  now  the  equi- 
librium is  destroyed  in  a  way  just  the  reverse  of  the  preceding 
cases.  The  water  between  the  balls  is  too  high  for  that  which  de- 
presses, and  too  low  for  that  which  raises  the  water,  so  that  the 
former  is  pushed  away,  and  the  latter  is  drawn  away. 

The  first  case,  which  is  by  far  the  most  common,  explains  the 
fact  often  observed,  that  floating  fragments  are  liable  to  be  gath- 


QUESTIONS    IN    HYDROSTATICS.  151 

ered  into  clusters ;  for  most  substances  are  capable  of  being  wet, 
and  therefore  they  raise  the  water  about  them. 

230.  Illustrations  of  Capillary  Action.— It  is  by  capil- 
lary action  that  a  part  of  the  water  which  falls  on  the  earth  is 
kept  near  its  surface,  instead  of  sinking  to  the  lowest  depths  of 
the  soil.    This  force  aids  the  ascent  of  sap  in  the  pores  of  plants. 
It  lifts  the  oil  between  the  fibres  of  the  lamp-wick  to  the  place  of 
combustion.     Cloth  rapidly  imbibes  moisture  by  its  numerous 
capillary  spaces,  so  that  it  can  be  used  for  wiping;  things  dry.    If 
paper  is  not  sized,  it  also  imbibes  moisture  quickly,  and  can  be 
used  as  blotting-paper  ;  but  when  its  pores  are  filled  with  sizing, 
to  fit  it  for  writing,  it  absorbs  moisture"  only  in  a  slight  degree, 
and  the  ink  which  is  applied  to  it  must  dry  by  evaporation. 

The  great  strength  of  the  capillary  force  is  shown  in  the  effects 
produced  by  the  swelling  of  wood  and  other  substances  when  kept 
wet.  Dry  wooden  wedges,  driven  into  a  groove  cut  around  a 
cylinder  of  stone,  and  then  occasionally  wet,  will  at  length  cause 
it  to  break  asunder.  As  the  pores  between  the  fibres  of  a  rope 
run  around  it  in  spiral  lines,  the  swelling  of  a  rope  caused  by 
keeping  it  wet  will  contract  its  length  with  immense  force. 

231.  Questions  in  Hydrostatics. — 

1.  The  diameters  of  the  two  cylinders  of  a  hydraulic  pressure 
one  inch  and  one  foot •,  respectively;  before  the  piston  descends,  the 
column  of  water  in  the  small  cylinder  is  two  feet  higher  than  the 
bottom  of  the  large  piston.    Suppose  that  by  a  screw  a  force  of 
500  Ibs.  is  applied  to  the  .small  piston ;  what  is  the  whole  force  ex- 
erted on  the  large  piston  at  the  beginning  of  the  stroke  ? 

Ans.  72098.17  Ibs. 

2.  A  junk  bottle,  whose  lateral  surface  contained  50  square 
inches,  being  let  down  into  the  sea  3000  feet,  what  pressure  do 
the  sides  of  the  bottle  sustain,  no   allowance    being  made  for 
the  increased  specific  gravity  of  the  sea-water  ? 

Ans.  65104.166  Ibs. 

3.  A  Greenland  whale  sometimes  has  a  surface  of  3600  square 
feet ;  what  pressure  would  he  bear  at  the  depth  of  800  fathoms  ? 

Ans.  1080,000,000  Ibs.,  or  more  than  482,142  tons.  ' 

4.  A  mill-dam,  running  perpendicularly  across  a  river,  slopes 
at  an  angle  of  25  degrees  with  the  horizon.    The  average  depth 
of  the  stream  is  12  feet,  and  its  breadth  500  yards;  required  the 
amount  of  pressure  on  the  dam  ? 

Ans.  15,971,906  Ibs.,  or  7130  +  tons. 

5.  A  mineral  weighs  960  grains  in  air,  and  739  grains  in  water; 
what  is  its  specific  gravity  ?  Ans.  4.34-4. 


152  HYDROSTATICS. 

6.  What  are  the  respective  weights  of  two  equal  cubical  masses 
of  gold  and  cork,  each  measuring  2  feet  on  its  linear,  edge  ? 

Ans.  The  gold  weighs  9625  Ibs.  =  4.297  tons;  the  cork 
weighs  120  Ibs. 

7.  A  mass  of  granite  contains  5949  cubic  feet.    The  specific 
gravity  of  a  fragment  of  it  is  found  to  be  2.6 ;  what  does  the  mass 
weigh  ?  Ans.  431.568  tons. 

8.  An  island  of  ice  rises  30  feet  out  of  water,  and  its  upper 
surface  is  a  circular  plane,  containing  fths  of  an  acre.     On  the 
supposition  that  the  mass  is  cylindrical,  required  its  weight,  and 
depth  below  the  water,  the  specific  gravity  of  sea-water  being 
1.0263,  and  that  of  ice  .92. 

Ans.  Weight,  242,900  tons ;  depth,  259.64  feet. 

9.  Wishing  to  ascertain  the  exact  number  of  cubic  inches  in  a 
very  irregular  fragment  of  stone,  I  ascertained  its  loss  of  weight  in 
water  to  be  5.346  ounces;  required  its  volume. 

Ans.  9.238  cubic  inches. 

10.  Hiero,  king  of  Syracuse,  ordered  his  jeweller  to  make  him 
a  crown  of  gold  containing  63  ounces.     The  artist  attempted  a 
fraud  by  substituting  a  certain  portion  of  silver;  which  being  sus- 
pected, the  king  appointed  Archimedes  to  examine  it.    Archi- 
medes, putting  it  into  water,  found  it  raised  the  fluid  8.2245 
inches;  and  having  found  that  the  inch  of  gold  weighs  10.36 
ounces,  and  that  of  silver  5.85  ounces,  he  discovered  what  part  of 
the  king's  gold  had  been  purloined ;  it  is  required  to  repeat  the 
process.  Ans.  28.8  ounces. 

11.  The  specific  gravity  of  lead  being  11.35 ;  of  cork,  .24 ;  of 
fir,  .45 ;  how  much  cork  must  be  added  to  60  Ibs.  of  lead,  that  the 
united  mass  may  weigh  as  much  as  an  equal  bulk  of  fir  ? 

Ans.  65.8527  Ibs. 

12.  A  cone,  whose  specific  gravity  is  |,  floats  on  water  with  its 
vertex  downward ;  what  part  of  the  axis  is  immersed  ? 

Ans.  One-half. 

13.  A  cone,  having  the  same  specific  gravity  as  the  above, 
floats  with  its  vertex  upward ;  how  much  of  its  axis  is  immersed  ? 

Ans.  0.0436. 

.  14.  What  is  the  weight  of  a  chain  of  pure  gold;  which  raises 
the  water  1  inch  in  height,  in  a  cubical  vessel  whose  side  is  3 
inches  ?  and  suppose  a  chain  of  the  same  weight  were  adulterated 
with  14|  ounces  of  silver ;  how  much  higher  would  it  raise  the 
water  in  the  vessel  ?  1  ft.  water  =  911.458  oz.  troy. 

Ans.  Weight  =  91.35  oz. ;  height  .133  in.  more. 


VELOCITY    OF    DISCHARGE.  153 

CHAPTER   II. 

LIQUIDS    IN    MOTION. 

232.  Depth  and  Velocity  of  Discharge.—  From  an  aper- 
ture which  is  small,  compared  with  the  breadth  of  the  reservoir, 
the  velocity  of  discharge  varies  as  the  square  root  of  the  depth.  For 
the  pressure  on  a  given  area  varies  as  the  depth  (Art.  207).  If  the 
area  is  removed,  this  pressure  is  a  force  which  is  measured  by  the 
momentum  of  the  water  ;  therefore  the  momentum  varies  as  the 
depth  (d).  But  momentum  varies  as  the  mass  (q)  multiplied  by 
the  velocity  (v)  ;  hence  q  v  <x  d.  But  it  is  obvious  that  q  and  v 
vary  alike,  since  the  greater  the  velocity,  the  greater  in  the  same 

ratio  is  the  quantity  discharged.    Therefore,  (f  oc  d,  or  q  x  d~  ; 

also  v2  oc  d,  or  v  cc  d2. 

Not  only  does  the  velocity  vary  as  the  square  root  of  the  depth 
of  the  orifice,  but  it  is  equal  to  that  acquired  ~by  a  body  falling 
through  the  depth. 

Let  h  =  the  height  of  the  liquid  above  the  orifice,  and  h'  =  the 
height  of  an  infinitely  thin  layer  at  the  orifice. 

If  this  thin  layer  were  to  fall  through  the  height  h',  under  the 
action  of  its  own  weight  or  pressure,  the  velocity  acquired  would  be 
v'  =  V2gli'  (Art.  28). 

Denoting  the  velocity  generated  by  the  pressure  of  the  entire 
column  by  v,  we  have,  since  velocity  x  1/depth, 


v  = 


But  V%gh  is  also  the  velocity  acquired  in  falling  through  the 
distance  h  (Art.  28). 

From  an  orifice  IGy^  feet  below  the  surface  of  water,  the  veloc- 
ity of  discharge  is  32  £  feet  per  second,  because  this  is  the  velocity 
acquired  in  falling  16T^  feet;  and  at  a  depth  four  times  as  great, 
that  is,  64|  feet,  the  velocity  will  only  be  doubled,  that  is,  64]  feet 
per  second. 

As  the  velocity  of  discharge  at  any  depth  is  equal  to  that  of  a 
body  which  has  fallen  a  distance  equal  to  the  depth,  it  is  theoreti- 
cally immaterial  whether  water  is  taken  upon  a  wheel  from  a  gate 
at  the  same  level,  or  allowed  to  fall  on  the  wheel  from  the  top  of 
the  reservoir.  In  practice,  however,  the  former  is  best,  on  ac- 


154  HYDROSTATICS. 

count  of  the  resistance  which  water  meets  with  in  falling  through 
the  air. 

233.  Descent  of  Surface. — When  water  is  discharged  from 
the  bottom  of  a  cylindric  or  prismatic  vessel,  the  surface  descends 
with  a  uniformly  retarded  motion.    For  the  velocity  with  which 
the  surface  descends  varies  as  the  velocity  of  the  stream,  and  there- 
fore as  the  square  root  of  the  depth  (Art.  232).    But  this  is  a 
characteristic  of  uniformly  retarded  motion,  that  it  varies  as  the 
square  root  of  the  distance  from  the  point  where  the  motion  ter- 
minates, as  in  the  case  of  a  body  ascending  perpendicularly  from 
the  earth. 

The  descent  of  the  surface  of  water  in  a  prismatic  vessel  has 
been  used  for  measuring  time.  The  clepsydra,  or  water-clock  of 
the  Eomans,  was  a  time-keeper  of  this  description.  The  gradua- 
tion must  increase  upward,  as  the  odd  numbers  1,  3,  5,  7,  &c. ; 
since,  by  the  law  of  this  kind  of  motion,  the  spaces  passed  over  in 
equal  times  are  as  those  numbers. 

If  a  prismatic  vessel  is  kept  full,  it  discharges  tivice  as  much 
water  in  the  same  time  as  if  it  is  allowed  to  empty  itself.  For  the 
.velocity,  in  the  first  instance,  is  uniform;  and  in  the  second  it  is 
uniformly  retarded,  till  it  becomes  zero.  We  reason  in  this  case, 
therefore,  as  in  regard  to  bodies  moving  uniformly,  and  with  mo- 
tion uniformly  accelerated  from  rest,  or  uniformly  retarded  till  it 
ceases  (Art.  25),  that  the  former  motion  is  twice  as  great  as  the 
latter. 

234.  Discharge  from  Orifices  in  Different  Situations.— 

Other  circumstances  besides  area  and  depth  of  the  aperture  are 
found  to  have  considerable  influence  on  the  velocity  of  discharge. 
Observations  on  the  directions  of  the  filaments  are  made  by  intro- 
ducing into  the  water  particles  of  some  opaque  substance,  having 
the  same  density  as  water,  whose  movements  are  visible.  From 
such  observations  it  appears  that  the  particles  of  water  descend  in 
vertical  lines,  until  they  arrive  within  three  or  four  inches  of  the 
aperture,  when  they  gradually  turn  in  a  direction  more  or  less 
oblique  toward  the  place  of  discharge.  This  convergence  of  the 
filaments  extends  outside  of  the  vessel,  and  causes  the  stream  to 
diminish  for  a  short  distance,  and  then  increase.  The  smallest 
section  of  the  stream,  called  the  vena  contracta,  is  at  a  distance 
from  the  aperture  varying  from  one-half  of  its  diameter  to  the 
whole. 

If  water  is  discharged  through  a  circular  aperture  in  a  thin 
plate  in  the  bottom  of  the  reservoir,  and  at  a  distance  from  the 
sides,  as  in  Fig.  161,  1,  the  filaments  form  the  vena  contracts  at  a 
distance  beyond  the  aperture  equal  to  one-half  of  its  diameter;  the 


FRICTION    IN    PIPES. 


155 


area  of  the  section  at  the  vena  contracta  is  less  than  two-thirds 
(0.64)  of  the  area  of  the  aperture ;  and  the  quantity  discharged  is 
also  about  two-thirds  of  that  obtained  by  calculation  for  the  full 
size  of  the  aperture. 

FIG.  161. 


If  the  reservoir  terminates  in  a  short  pipe  or  ajutage,  whose 
interior  is  adapted  to  the  curvature  of  the  filaments,  as  far  as  to 
the  vena  contracta,  or  a  little  beyond,  as  in  Fig.  161,  2,  it  is  found 
the  most  favorable  for  free  discharge,  which  in  some  cases  reaches 
0.98  of  the  theoretical  discharge.  The  stream  is  smooth  and  pel- 
lucid like  a  rod  of  glass.  The  most  unfavorable  form  is  that  in 
which  the  ajutage,  instead  of  being  external,  as  in  the  case  just 
described,  projects  inward,  as  in  Fig.  161,  3 ;  the  filaments  in  this 
case  reach  the  aperture,  some  ascending,  others  descending,  and 
therefore  interfere  with  each  other.  Hence  the  stream  is  much 
roughened  in  its  appearance,  and  the  flow  is  only  0.53  of  what  is 
due  to  the  size  of  the  aperture  and  its  depth. 

"When  the  aperture  is  through  a  thin  plate,  the  contraction  of 
the  stream  and  the  amount  of  discharge  are  both  modified  by  the 
circumstance  of  being  near  one  or  more  sides  of  the  reservoir. 
There  'is  little  or  no  contraction  on  the  side  next  the  wall  of  the 
vessel,  since  the  filaments  have  no  obliquity  on  that  side ;  and  the 
quantity  is  on  that  account  increased.  The  filaments  from  the 
opposite  side  also  divert  the  stream  a  few  degrees  from  the  perpen- 
dicular (Fig.  161,  4). 

235.  Friction  in  Pipes. — As  has.  just  been  stated,  an  ajutage 
extending  to  or  slightly  beyond  the  vena  contracta,  and  adapted 
to  the  form  of  the  stream,  very  much  increases  the  quantity  dis- 
charged ;  but  beyond  that,  the  longer  the  pipe,  the  more  does  it 
impede  the  discharge  by  friction.  The  friction  varies  directly  as 
the  length  of  the  pipe,  and  inversely  as  its  diameter.  In  order, 
therefore,  to  convey  water  at  a  given  rate  through  a  long  pipe,  it  is 
necessary  either  to  increase  the  head  of  water  or  to  enlarge  the 
pipe,  so  as  to  compensate  for  friction.  If  a  given  quantity  of 


156 


HYDROSTATICS. 


water  is  discharged  per  second  by  an  aperture  five  inches  in  diam- 
eter, through  the  thin  bottom  of  a  reservoir,  and  we  wish  to  dis- 
charge the  same  at  the  end  of  a  horizontal  pipe  150  feet  long,  and 
of  the  same  diameter  as  the  orifice,  it  will  require  ten  times  the 
head  of  water  to  accomplish  it;  or  it  may  be  done  by  the  same 
head  of  water,  if  we  use  a  pipe  about  8  inches,  instead  of  5  inches, 
in  diameter. 

An  aqueduct  should  be  as  straight  as  possible,  not  only  to 
avoid  unnecessary  increase  of  length,  but  because  the  force  of  the 
stream  is  diminished  by  all  changes  of  direction.  If  there  must  be 
change,  it  should  be  a  gradual  curve,  and  not  an  abrupt  turn. 
"When  a  pipe  changes  its  direction  by  an  angle,  instead  of  a  curve, 
there  is  a  useless  expenditure  of  force ;  a  change  of  90°  requires 
that  the  head  of  water  should  be  increased  by  nearly  the  height 
due  to  the  velocity  of  discharge.  For  instance,  if  the  discharge  is 
eight  feet  per  second  (which  is  the  velocity  due  to  one  foot  of  fall), 
then  a  right  angle  in  the  pipe  requires  that  the  head  of  water 
should  be  increased  by  nearly  one  foot,  in  order  to  maintain  that 
velocity. 

236.  Jets. — Since  a  body,  when  projected  upward  with  a  cer- 
tain velocity,  will  rise  to  the  same  height  as  that  from  which  it 
must  have  fallen  to  acquire  that  velocity,  therefore,  if  water  issue 
from  the  side  of  a  vessel  through  a  pipe  bent  upward,  it  would, 
were  it  not  for  the  resistance  of  the  air  and  friction  at  the  orifice, 
rise  to  the  level  of  the  water  in  the  reservoir.  If  water  is  dis- 
charged from  an  orifice  in  any  other  than  a  vertical  direction,  it 
describes  a  parabola,  since  each  particle  may  be  regarded  as  a  pro>- 
jectile  (Art.  44). 

If  a  semicircle  be  described  on  the  perpendicular  side  of  a 
vessel  as  a  diameter,  and  water 
issue  horizontally  from  any  point, 
its  range,  measured  on  the  level  of 
the  base,  equals  twice  the  ordinate 
of  that  point.  For,  the  velocity 
with  which  the  fluid  issues  from 
the  vessel,  being  that  which  is  due 
to  the  height  B  G  (Fig.  162),  is 
V2j  .  B  G  (Art.  28).  But  after 
leaving  the  orifice,  it  arrives  at  the 


C  J)  JS 

horizontal  plane  in  the  time  in  which  a  body  would  fall 
through  G  D,  which  is  y 


GD 


F 

freely 

Since  the  horizontal  motion 


is  uniform,  the  space  equals  the  product  of  the  time  by  the  veloc- 
ity; that  is,  D  E  =  ]/  -    —  x  V%  g  .  B  G.=  2 

\j 


G  .  GD  = 


RIVERS.  157 

2  G  H,  or  twice  the  ordinate  of  the  semicircle  at  the  place  of  dis- 
charge. 

The  greatest  range  occurs  when  the  fluid  issues  from,  the 
centre,  for  then  the  ordinate  is  greatest ;  and  the  range  at  equal 
distances  above  and  below  the  centre  is  the  same. 

The  remarks  already  made  respecting  pipes  apply  to  those 
which  convey  water  to  the  jets  of  fire-engines  and  fountains.  If 
the  pipe  or  hose  is  very  long,  or  narrow,  or  crooked,  or  if  the  jet- 
pipe  is  not  smoothly  tapered  from  the  full  diameter  of  the  hose  to 
the  aperture,  much  force  is  lost  by  friction  and  other  resistances, 
especially  in  great  velocities.  If  the  length  of  hose  is  even  twenty 
times  as  great  as  its  diameter,  32  per  cent,  of  height  is  lost  in  the 
jet,  and  more  still  when  the  ratio  of  length  to  diameter  is  greater 
than  this. 

237.  Rivers. — Friction  and  change  of  direction  have  great 
influence  on  the  flow  of  rivers.  A  dynamical  equilibrium,  as  it  is 
called,  exists  between  gravity,  which  causes  the  descent,  and  the 
resistances,  which  prevent  acceleration,  beyond  a  certain  moderate 
limit ;  so  that,  in  general,  the  water  of  a  river  moves  uniformly. 
The  velocity  in  all  parts  of  the  same  section,  however,  is  not  the 
same;  it  is  greatest  at  that  part  of  the  surface  where  the  depth  is 
greatest,  and  least  in  contact  with  the  bed  of  the  stream. 

To  find  the  mean  velocity  through  a  given  section,  it  is  neces- 
sary to  float  bodies  at  various  places  on  the  surface,  and  also  below 
it,  to  the  bottom,  and  to  divide  the  sum  of  all  the  velocities  thus 
obtained,  by  the  number  of  observations.  To  obtain  the  quantity 
of  water  which  flows  through  a  given  section  of  a  river,  having 
determined  the  velocity  as  above,  find  next  the  area  of  the  section, 
by  taking  the  depth  at  various  points  of  it,  and  multiplying  the 
mean  depth  by  the  breadth.  The  quantity  of  water  is  then  found 
by  multiplying  the  area  by  the  velocity. 

The  increased  velocity  of  a  stream  during  a  freshet,  while  the 
stream  is  confined  within  its  banks,  exhibits  something  of  the  ac- 
celeration which  belongs  to  bodies  descending  on  an  inclined 
plane.  It  presents  the  case  of  a  river  flowing  upon  the  top  of 
another  river,  and  consequently  meeting  with  much  less  resistance 
than  when  it  runs  upon  the  rough  surface  of  the  earth  itself.  The 
augmented  force  of  a  stream  in  a  freshet  arises  from  the  simulta- 
neous increase  of  the  quantity  of  water  and  the  velocity.  In  con- 
sequence of  the  friction  of  the  banks  and  beds  of  rivers,  and  the 
numerous  obstacles  they  meet  with  in  their  winding  course,  their 
velocity  is  usually  very  small,  not  more  than  three  or  four  miles 
per  hour ;  whereas,  were  it  not  for  these  impediments,  it  would 
become  immensely  great,  and  its  effects  would  be  exceedingly  dis- 


158 


HYDROSTATICS. 


astrous.  A  very  slight  declivity  is  sufficient  for  giving  the  run- 
ning motion  to  water.  The  largest  rivers  in  the  world  fall  about 
five  or  six  inches  in  a  mile. 

238.  Hydraulic   Pumps.— The  most  common  pumps  for 
raising  water  operate  on  a  principle  of  pneumatics,  and  will  be  de- 
scribed under  that  subject. 

In  the  lifting-pump  the  water  is  pushed  up  in  the  pump  tube 
by  a  piston  placed  below  the  water-level.  In  the  tube  A  B  (Fig. 
163)  is  a  fixed  valve  V,  a  little  below  the 
water-level  L  L,  while  still  lower  is  the  pis- 
ton P,  in  which  there  is  a  valve.  Both  of 
these  valves  open  upward.  The  •  piston  is 
attached  to  a  rod,  which  extends  downward 
to  the  frame  F  F.  This  frame  can  be  moved 
up  and  down  on  the  outside  of  the  tube  by  a 
lever.  When  the  piston  descends,  the  water 
passes  through  its  valve  by  hydrostatic  press- 
ure; and  when  raised,  it  pushes  the  water 
before  it  through  the  fixed  valve,  which  then 
prevents  its  return.  In  this  manner,  by  re- 
peated strokes,  the  water  can  be  driven  to 
any  height  which  the  instrument  can  bear. 

The  chain  pump  consists  of  an  endless 
chain  with  circular  disks  attached  to  it  at 
intervals  of  a  few  inches,  which  raise  the 
water  before  them  in  a  tube,  by  means  of  a 
wheel  over  which  the  chain    passes;    the 
wheel  may  be  turned  by  a  crank.     The 
disks  cannot  fit  closely  in  the  tube  without  causing  too  great  re- 
sistance; hence,  a  certain  velocity  is  requisite  in  order  to  raise 
water  to  the  place  of  discharge, ;  and  after  the  working  of  the 
pump  ceases,  the  water  soon  descends  to  the  level  in  the  well. 

239.  The   Hydraulic   Ram. — When  a  large  quantity  of 
water  is  descending  through  an  inclined  pipe,  if  the  lower  extrem- 
ity is  suddenly  closed,  since  water  is  nearly  incompressible,  the 
shock  of  the  whole  column  is  received  in  a  single  instant,  and  if 
no  escape  is  provided,  is  very  likely  to  burst  the  pipe.    The  inten- 
sity of  the  shock  of  water  when  stopped  is  made  the  means  of 
raising  a  portion  of  it  above  the  level  of  the  head.    The  instru- 
ment for  effecting  this  is  called  the  hydraulic  ram.    At  the  lower 
end  of  a  long  pipe,  P  (Fig.  164),  is  a  valve,  F,  opening  downward ; 
near  it,  another  valve,  F',  opens  into  the  air-vessel,  A;  and  from 
this  ascends  the  pipe,  T,  in  which  the  water  is  to  be  raised.    As 


WATER-WHEELS. 


159 


the  valve  F  lies  open  by  its  weight,  the  water  runs  out,  till  its 
momentum  at  length  shuts  it,  and  the  entire  column  is  suddenly 

FIG.  164. 


FIG.  1G5. 


stopped;  this  impulse  forces  the  water  into  the  air-vessel,  and 
thence,  by  the  compressed  air,  up  the  tube  T.  As  soon  as  the 
momentum  is  expended,  the  valve  V  drops,  and  the  process  is 
repeated. 

240.  Water- Wheels  with  a  Horizontal  Axis.— The  over- 
shot  ivfieel  (Fig.  165)  is  construct- 
ed with  buckets  on  the  circumfer- 
ence, which  receive  the  water  just 
after  passing  the  highest  point, 
and  empty  themselves  before 
reaching  the  bottom.  The  weight 
of  the  water,  as  it  is  all  on  one 
side  of  a  vertical  diameter,  causes 
the  wheel  to  revolve.  It  is  usual- 
ly made  as  large  as  the  fall  will 
allow,  and  will  carry  machinery 
with  a  very  small  supply  of  water, 
if  the  fall  is  only  considerable. 
The  moment  of  each  bucket-full 
constantly  increases  from  a,  where 
it  is  filled,  to  F,  where  its  acting  distance  is  radius,  and  therefore 
a  maximum.  From  F  downward  the  moment  decreases,  hoth  by 
loss  of  water  and  diminution  of  acting  distance,  and  becomes  zero 
at  L. 

The  undershot  wheel  (Fig.  166)  is  re-  FIG.  166. 

volved  by  the  momentum  of  running  water, 
which  strikes  the  float-boards  on  the  lower 
side.  WJien  these  are  placed,  as  in  the 
figure,  perpendicular  to  the  circumference, 
the  wheel  may  turn  either  way ;  this  is  the 
construction  adopted  in  tide-mills.  When 
the  wheel  is  required  to  turn  only  in  one 
direction,  an  advantage  is  gained  by  placing 


ICO 


HYDROSTATICS. 


FIG.  167. 


the  floa  -boards  so  as  to  present  an  acute  angle  toward  the  current? 
by  which  means  the  water  acts  partly  by  its  weight,  as  in  the  over- 
shot wheel.  The  undershot  wheel  is  adapted  to  situations  where 
the  supply  of  water  is  always  abundant. 

In  the  breast  wheel  (Fig.  167)  the  water  is  received  upon  the 
float-boards  at  about  the  height  of  the  axis,  and  acts  partly  by  its 
weight,  and  partly  by  its  mo- 
mentum. The  planes  of  the 
float-boards  are  set  at  right 
angles  to  the  circumference  of 
the  wheel,  and  are  brought  so 
near  the  mill-course  that  the 
water  is  held  and  acts  by  its 
weight,  as  in  buckets. 

241.  The  Turbine.— This 

very  efficient  water-wheel,  fre- 
quently called  the  French  tur- 
bine, is  of  modern  invention, 
and  has  received  its  chief  im- 
provements in  this  country.  It  revolves  on  a  vertical  axis,  and 
surrounds  the  bottom  of  the  reservoir  from  which  it  receives  the 
water.  The  lower  part  of  the  reservoir  is  divided  into  a  large 
number  of  sluices  by  curved  partitions,  which  direct  the  water 
nearly  into  the  line  of  a  tangent,  as  it  issues  upon  the  wheel.  The 
vanes  of  the  wheel  are  curved  in  the  opposite  direction,  so  as  to 
receive  the  force  of  the  issuing  streams  at  right  angles.  The  hori- 
zontal section  (Fig.  168)  shows 
the  lower  part  of  the  reservoir 
with  its  curved  guides,  a,  a,  a, 
and  the  wheel  with  its  curved 
vanes,  v,  v,  v,  surrounding  the 
reservoir ;  D  is  the  central  tube, 
through  which  the  axis  of  the 
wheel  passes.  Fig.  169  is  a  ver- 
tical section  of  the  turbine ;  but 
it  does  not  present  the  guides 
of  the  reservoir,  nor  the  vanes 
of  the  wheel.  C  G,  C  G,  is  the 
outer  wall  of  the  reservoir; 
D,  D,  its  inner  wall  or  tube ; 
F  F,  the  base,  curved  so  as  to 

turn  the  descending  water  gradually  into  a  horizontal  direction. 

The  outer  wall,  which  terminates  at  G  G,  is  connected  with  the 

•  base  and  tube  by  the  guides  which  are  shown  at  a,  a,  in  Fig,  168, 


FIG.  168. 


THE    TURBINE. 


1C1 


The  lower  rim  of  the  wheel,  //,  //,  is  connected  with  the  upper 
rim,  P,  P}  by  the  vanes  between  them,  v,  v  (Fig.  168),  and  to  the 

FIG.  169. 


axis,  Et  E,  by  the  spokes  /,  /.  The  gate,  J,  J,  is  a  thin  cylinder 
which  is  raised  or  lowered  between  the  wheel  and  the  sluices  of 
the  reservoir.  The  bottom  of  the  axis  revolves  in  the  socket  K, 
and  the  top  connects  with  the  machinery.  As  the  reservoir  can- 
not be  supported  from  below,  it  is  suspended  by  flanges  on  the 
masonry  of  the  wheel-pit,  or  on  pillars  outside  of  the  wheel. 
To  prevent  confusion  in  the  figure,  the  supports  of  the  reservoir 
and  the  machinery  for  raising  the  gate  are  omitted.  By  the 
curved  base  and  guides  of  the  reservoir,  the  water  is  conducted  in 
a  spiral  course  to  the  wheel,  with  no  sudden  change  of  direction, 
and  thus  loses  very  little  of  its  force.  The  wheel  usually  runs 
below  the  level  of  the  water  in  the  wheel-pit,  as  represented  in 
the  figure,  L  L  being  the  surface  of  the  water.  The  reservoir  is 
sometimes  merely  the  extremity  of  a  large  tapering  tube  or  supply 
pipe,  bent  from  a  horizontal  to  a  vertical  direction.  In  such  a 
case,  the  tube  D  D,  in  which  the  axis  runs,  passes  through  the 
upper  side  of  the  supply  pipe.  The  figure  represents  only  the 
lower  part. 

242.  Barker's  Mill. — This  machine  operates  on  the  principle 
of  unbalanced  hydrostatic  pressure.  It  consists  of  a  vertical  hollow 
cylinder,  A  B  (Fig.  170),  free  to  revolve  on  its  axis  M  N,  and 
having  a  horizontal  tube  connected  with  it  at  the  bottom.  Near 
11 


162 


HYDROSTATICS. 


FIG   170. 


each  end  of  the  horizontal  tube,  at  P  and  P',  is  an  orifice,  one  on 
one  side,  and  one  on  the  opposite.  The 
cylinder,  being  kept  full  of  water,  whirls 
in  a  direction  opposite  to  that  of  the  dis- 
charging streams  from  P  and  P'.  This 
is  owing  to  the  fact  that  hydrostatic 
pressure  is  removed  from  the  apertures, 
while  on  the  interior  of  the  tube,  at 
points  exactly  opposite  to  them,  are 
pressures  which  are  now  unbalanced, 
but  which  would  be  counteracted  by 
the  pressures  at  the  apertures,  if  they 
were  closed.  The  centrifugal  force, 
after  the  machine  is  in  rotation,  has 
the  effect  to  increase  the  pressure,  and 
therefore  the  speed  of  rotation. 

243.  Resistance  to  Motion  in  a  Liquid. — The  resistance 

which  a  body  encounters  in  moving  through  any  fluid  arises  from 
the  inertia  of  the  particles  of  the  fluid,  their  want  of  perfect  mo- 
bility among  each  other,  and  friction.  Only  the  first  of  these 
admits  of  theoretical  determination.  So  far  as  the  inertia  of  the 
fluid  is  concerned,  the  resistance  which  a  surface  meets  with  in 
moving  perpendicularly  through  it  varies  as  the  square  of  the  ve- 
locity. For  the  resistance  is  measured  by  the  momentum  imparted 
by  the  moving  body  to  the  fluid.  And  this  momentum  (m)  varies 
as  the  product  of  the  quantity  of  fluid  set  in  motion  (q\  and  its 
velocity  (v) ;  or  m  cc  q  v.  But  it  is  obvious  that  the  quantity  dis- 
placed varies  as  the  velocity  of  the  body,  or  q  oc  v ;  hence  m  oc  v*. 
Therefore  the  resistance  varies  as  the  square  of  the  velocity. 

This  proposition  is  found  to  hold  good  in  practice,  where  the 
velocity  is  small,  as  the  motions  of  boats  or  ships  in  water;  bat 
when  the  velocity  becomes  very  great,  as  that  of  a  cannon  ball, 
the  resistance  increases  in  a  much  higher  ratio  than  as  the  square 
of  the  velocity.  Since  action  and  reaction  are  equal,  it  makes  no 
difference,  in  the  foregoing  proposition,  whether  we  consider  the 
body  in  motion  and  the  fluid  at  rest,  or  the  fluid  in  motion  and 
striking  against  the  body  at  rest. 

Since  the  resistance  increases  so  rapidly,  there  is  a  wasteful 
expenditure  of  force  in  trying  to  attain  great  velocities  in  naviga- 
tion. For,  in  order  to  double  the  velocity  of  a  steamboat,  the 
force  of  the  steam  must  be  increased  four  fold ;  and  in  order  to 
triple  its  velocity,  the  force  must  become  nine  times  as  great. 

When  the  resistance  becomes  equal  to  the  moving  force,  the 
body  moves  uniformly,  and  is  said  to  be  in  a  state  of  dynamical 


WATER    WAVES. 


163 


equilibrium.  Thus,  a  body  falling  freely  through  the  air  by 
gravity  does  not  continue  to  be  accelerated  beyond  a  certain 
limit,  but  is  finally  brought,  by  the  resistance  of  the  air,  to  a  uni- 
form motion. 

244.  Water  Waves. — These  are  moving  elevations  of  water, 
caused  by  some  force  which  acts  unequally  on  its  surface.    There 
are  two  very  different  kinds  of  waves,  called,  respectively,  waves 
of  oscillation  and  waves  of  translation.    In  the  first  kind  the  par- 
ticles of  water  have  a  vibratory  or  reciprocating  motion,  by  which 
the  vertical  columns  are  alternately  lengthened  and  shortened.    A 
familiar  example  of  this  kind  is  the  sea-wave.    In  the  waves  of 
translation  the  particles  are  raised,  transferred  forward,  and  then 
deposited  in  a  new  place,  without  any  vibratory  movement. 

245.  Waves  of  Oscillation. — If  a  pebble  be  tossed  upon 
still  water,  it  crowds  aside  the  particles  beneath  it,  and  raises  them 
above  the  level,  forming  a  wave  around  it  in  the  shape  of  a  ring. 
As  soon  as  this  ring  begins  to  descend,  it  elevates  above  the  level 
another  portion  around  itself,  and  thus  the  ring-wave  continues 
to  spread  outward  every  way  from  the  centre.    But  in  the  mean- 
time the  water  at  the  centre,  as  it  rises  toward  the  level,  acquires 
a  momentum  which  lifts  it  above  that  level.    From  that  position  it 
descends,  and  once  more  passes  below  the  level,  thus  starting  a 
new  wave  around  it,  as  at  first,  only  of  less  height.    Hence,  we  see 
as  the  result  of  the  first  disturbance,  a  series  of  concentric  waves 
continually  spreading  outward  and  di- 
minishing   in  height  at  greater    dis- 
tances, until  they  cease  to  be  visible. 

In  Fig.  171  are  represented  three  circu- 
lar waves  at  one  of  the  moments  of 
time  when  the  centre  is  lowest.  The 
shaded  parts  are  the  basins  or  troughs, 
and  the  light  parts,  c,  c,  c,  are  the  ridges 
or  crests.  Fig.  172  is  a  vertical  section 
along  the  line  c,  c,  through  the  centre 
of  the  system,  corresponding  to  the  mo- 
mentary arrangement  of  Fig.  171.  The  central  basin  is  at  Z>,  and 
the  crests  at  c,  c,  c.  A  little  later,  when  either  crest  has  moved 
half  way  to  the  place  of  the  next  one, 
both  figures  will  have  become  reversed ; 
the  centre  will  be  a  hillock,  the  troughs 
will  be  at  c,  c,  and  the  crests  at  the 
middle  points  between  them. 

Except  in  the  circular  arrangement  of  the  crests  and  troughs 
around  a  centre,  the  waves  of  the  foregoing  experiment  illustrate 


FIG.  171. 


FIG.  172. 


1G4 


HYDROSTATICS. 


common  sea- waves.  They  constitute  a  system  of  elevations  and 
depressions  moving  along  the  surface  at  right  angles  to  the  line 
of  the  wave-crest. 

246.  Phases. — In  the  cross-section  (Fig.  172),  where   the 
waves  are  shown  in  profile,  any  particular  part  of  the  curve  is 
called  a  phase.    Different  phases  are  generally  unlike,  both  in  ele- 
vation and  in  movement.    The  corresponding  parts  of  different 
waves  are  called  like  phases ;  and  those  points  in  which  the  mo- 
lecular motions  are  reversed  are  called  opposite  phases.      The 
highest,  points  of  the  crests  of  two  waves  are  like  phases;  the 
highest  point  of  the  crest  and  the  lowest  point  of  the  trough  are 
opposite  phases.    Two  points  half  way  from  crest  to  trough,  one 
on  the  front  of  the  wave,  and  the  other  on  the  rear  of  it,  are  also 
opposite  phases,  although  they  are  at  the  same  elevation ;  for  they 
are  moving  in  opposite  directions.     The  length  of  a  wave  is  the 
horizontal  distance  between  two  successive  like  phases. 

247.  Molecular  Movements.— The  water  which  constitutes 
a  system  of  waves  does  not  advance  along  the  surface,  as  the  waves 
themselves  do ;  for  a  floating  body  is  not  borne  along  by  them,  but 
alternately  rises  and  falls  as  the  waves  pass  under  it.    Each  par- 
ticle of  water,  instead  of  advancing  with  the  wave,  oscillates  about 
its  mean  place,  alternately  rising  as  high  as  the  crest,  and  falling 
as  low  as  the  trough.    Its  path  is  the  circumference  of  a  vertical 
circle.    Let  B  Br  (Fig.  173)  represent  two  successive  troughs,  and 

FTG.  173. 


"a        b        c       d        e       f  '    h        a'       b' 


O  the  intervening  crest;  and  for  convenience  suppose  a  a,  the 
wave  length,  to  be  divided  into  eight  equal  parts.  The  waves 
move  in  the  direction  of  the  straight  arrow,  while  the  particles  of 
water  revolve  in  the  direction  of  the  bent  arrows.  The  points 
1,  2,  3,  &c.,  represent  particles  which,  if  the  water  were  at  rest, 
would  be  directly  above  the  points  «,  5,  c,  &c.  At  the  moment 
represented,  1  is  at  the  extreme  left  of  its  revolution,  2  is  at  45° 
below,  3  at  the  lowest  point,  &c.  When  the  wave  has  advanced 
one-eighth  of  its  length,  1  will  have  ascended  45°,  2  will  have  as- 
cended to  the  extreme  left,  and  each  of  the  eight  particles  will 


FORM    OF   WAVES.  165 

have  revolved  one-eighth  of  the  circumference  shown  in  the  figure. 
Then  4  will  be  at  the  bottom,  and  8  at  the  top.  Each  particle  of 
water  on  the  front  of  the  wave,  from  1  to  3,  and  from  7  to  3',  is 
ascending ;  each  one  on  the  rear,  from  3  to  7,  is  descending.  It 
is  plain  that  while  the  wave  advances  its  whole  length,  that  is, 
while  the  phase  B  is  moving  to  J9',  each  particle  makes  a  complete 
revolution;  3',  which  is  now  lowest,  will  be  lowest  again,  having 
in  the  meantime  occupied  all  other  points  of  the  circumference. 

Particles  below  the  surface,  as  far  as  the  wave  disturbance 
reaches,  perform  synchronous  revolutions,  but  in  smaller  circles, 
as  represented  in  the  figure. 

248.  Form  of  Waves  of  Oscillation.— The  sectional  form 
of  these  waves  is  that  of  the  inverted  trochoid,  a  curve  described 
by  a  point  in  a  circle  as  it  rolls  on  a  straight  line.    The  curvature 
of  the  crest  is  always  greater  than  that  of  the  trough,  and  the 
summit  may  possibly  be  a  sharp  ridge,  in  which  case  the  section 
of  the  trough  is  a  cycloid,  the  describing  point  of  the  rolling  circle 
being  on  the  circumference;  the  height  of  such  waves  is  to  their 
length  as  the  diameter  of  a  circle  to  the  circumference.    If  waves 
are  ever  higher  than  about  one-third  of  their  length,  the  summits 
are  broken  into  spray. 

249.  Distortion  of  the  Vertical  Columns.— Where  the 

surface  is  depressed  below  its  level,  some  of  the  water  must  be 
crowded  laterally  out  of  its  place,  and  the  vertical  columns,  being 
shorter,  must  necessarily  be  wider,  at  least  in  the  upper  part.  So, 
too,  where  the  surface  is  raised  above  its  level,  the  lengthened 
columns  must  be  narrower.  In  Fig.  173  these  effects  are  made 
apparent  as  the  necessary  result  of  the  revolutions  of  the  particles. 
The  dotted  lines,  1  «,  2  b,  3  c,  &c.,  were  all  vertical  lines  when  the 
water  was  at  rest.  But  now  they  are  swayed,  some  to  the  right 
and  some  to  the  left,  none  being  vertical,  except  under  the  highest 
and  lowest  points  of  the  waves.  Under  the  trough  the  lines  are 
spread  apart,  and  under  the  crest  they  are  drawn  together.  The 
sectional  figures  1  a  b  2,  2  b  c  3,  &c.,  which  would  all  be  rectangu- 
lar if  the  water  were  at  rest,  are  now  distorted  in  form,  the  upper 
parts  being  alternately  expanded  and  contracted  in  breadth  as  the 
successive  phases  pass  them. 

250.  Sea-Waves. — The  waves  raised  by  the  wind  rarely  ex- 
hibit the  precise  forms  above  described,  and  the  particles  rarely 
revolve  in  exact  circles,  partly  because  there  is  scarcely  eATer  a  sys- 
tem of  waves  undisturbed  by  other  systems,  which  are  passing 
over  the  water  at  the  same  time,  and  partly  because  the  wind, 
which  was  the  original  cause  of  the  waves,  acts  continually  upon 
their  surfaces  to  distort  and  confuse  them. 


166  HYDROSTATICS. 

The  interference  of  waves  denotes,  in  general,  the  resultant 
system,  which  is  produced  by  the  combination  of  two  or  more 
separate  systems.  The  joint  effect  of  two  systems  is  various,  ac- 
cording as  they  are  more  or  less  unlike  as  to  length  of  waves. 
But  even  if  two  systems  are  just  alike,  still  the  effect  of  interfer- 
ence will  vary,  according  to  the  coincidence  or  the  degree  of  dis- 
crepancy of  their  like  phases.  For  instance,  if  two  similar 
systems  exactly  coincide,  phase  for  phase,  the  waves  simply  have 
double  height;  or,  in  general  terms,  there  is  double  intensity  in 
the  wave  motion.  But  if  the  phases  of  one  system  exactly  coincide 
with  the  opposite  phases  of  the  other,  then  the  water  is  nearly 
level,  the  crests  of  each  system  filling  the  troughs  of  the  other. 
These  two  effects  may  be  plainly  seen  in  the  intersections  of  ring- 
waves  formed  by  dropping  two  pebbles  on  still  water. 

251.  Waves  of  Translation. — The  principal  characteristics 
of  the  wave  of  translation  are,  that  it  is  solitary — i.  e.,  it  does  not 
belong  to  a  system,  like  the  other  kind  ;  and  that  its  length  and 
velocity  both  depend  on  the  depth  of  the  water.  Where  the  water 
is  deeper,  the  wave  travels  faster,  and  its  length  (measured  in  the 
direction  of  its  progress)  is  longer.  A  wave  of  this  character  is 
started  in  a  canal  by  a  moving  boat;  and  when  the  boat  stops,  it 
moves  on  alone.  A  grand  example  of  this  species  is  found  in  the 
tide-wave  of  the  ocean.  It  is  called  the  wave  of  translation  be- 
cause the  particles  of  water  are  borne  forward  a  certain  distance 
while  the  wave  is  passing,  and  then  remain  at  rest. 


PART   III. 


CHAPTER  I. 

PROPERTIES   OF   GASES.— INSTRUMENTS   FOR  INVESTIGATION. 

252.  Gases  Distinguished  from  Liquids.— The  property 
of  mobility  of  particles,  which  belongs  to  all  fluids,  is  more  re- 
markable in  gases  than  in  liquids. 

While  gaseous  substances  are  compressed  with  ease,  they  are 
always  ready  to  expand  and  occupy  more  space.  This  property, 
called  dilatability,  scarcely  belongs  to  liquids  at  all.  The  force 
which  gases  show  in  expanding  is  called  tension. 

253.  Change  of  Condition.— Liquids,  and  even  solids,  may 
be  changed  into  the  gaseous  or  aeriform  condition  by  heating 
them  sufficiently.    By  being  cooled,  they  return  again  to  their 
former  state.    In  the  gaseous  form  they  are  called  vapors.    And 
some  substances  which  are  ordinarily  gases  can  be  so  far  cooled, 
especially  under  great  pressure,  as  to  be  reduced  to  the  liquid  or 
solid  form.    Those  which  have  never  been  thus  reduced  are  called 
permanent  gases. 

The  mechanical  properties  of  the  gases  may  all  be  illustrated 
by  experiments  performed  upon  atmospheric  air. 

254.  Mariotte's  Law.— 

At  a  given  temperature,  the  volume  of  air  is  inversely  as  the 
compressing  force. 

An  instrument  constructed  for  showing  this  is  called  Mariotte's 
tube.  The  end  B  (Fig.  174)  is  sealed,  and  A  open.  Pour  in  small 
quantities  of  mercury,  inclining  the  tube  so  as  to  let  air  in  or  out, 
till  both  branches  are  filled  to  the  zero  point.  The  air  in  the 
short  branch  now  has  the  same  tension  as  the  external  air,  since 
they  just  balance  each  other.  If  mercury  be  poured  in  till  th3 
column  in  the  short  tube  rises  to  C,  the  inclosed  air  is  reduced  to 
one-half  of  its  original  volume,  and  the  column  A  in  the  long  branch 
is  found  to  be  29  or  30  inches  above  the  level  of  C,  according  to 


168 


PNEUMATICS. 


174. 


the  barometer  at  the  time.  Thus,  two  atmospheres,  one  of  mer- 
cury, the  other  of  air  above  it,  have  compressed  the  inclosed  air 
into  one-half  its  volume.  If  the  tube  is  of 
sufficient  length,  let  mercury  be  poured  in 
again,  till  the  air  is  compressed  to  one-third 
of  its  original  space;  the  long  column, 
measured  from  the  level  of  the  mercury  in 
the  short  one,  is  now  twice  as  high  as  be- 
fore ;  that  is,  tliree  atmospheres,  two  of  mer- 
cury and  one  of  air,  have  reduced  the  same 
quantity  of  air  to  one-third  of  its  first  vol- 
ume. This  law  has  been  found  to  hold  good 
in  regard  to  atmospheric  air  up  to  a  pressure 
of  nearly  thirty  atmospheres. 

On  the  other  hand,  if  the  pressure  on  a 
given  mass  of  air  is  diminished,  its  volume 
is  found  to  increase  according  to  the  same 
law.  When  the  pressure  is  half  an  atmo- 
sphere, the  volume  is  doubled ;  when  one- 
third  of  an  atmosphere,  the  volume  is  three 
times  as  great,  &c. 

This  law  is  found,  however,  not  to  be 
strictly  applicable  to  all  the  gases.  Some 
are  compressed  a  little  more,  and  others  a 
little  less,  than  Mariotte's  law  would  re- 
quire. 

Since  the  tension  of  the  inclosed  air 
always  Jbalances  the  compressing  force,  and 
since  the  density  is  inversely  as  the  volume, 
it  follows  from  Mariotte's  law  that  when  the  temperature  is  the  same, 

The  tension  of  air  varies  as  the  compressing/ force  j  and 

TJie  tension  of  air  varies  as  its  density. 

255.  The  Air-Pump, — This  is  an  instrument  by  which 
nearly  all  the  air  can  be  removed  from  a  vessel  or  receiver.  It 
has  a  variety  of  forms,  one  of  which  is  shown  in  Fig.  175.  In  the 
barrel  B  an  air-tight  piston  is  alternately  raised  and  depressed  by 
the  lever,  the  piston-rod  being  kept  vertical  by  means  of  a  guide. 
The  pipe  P  connects  the  bottom  of  the  barrel  with  the  brass  plate 
//,  on  which  rests  the  receiver  R.  The  surface  of  the  plate  and 
the  edge  of  the  receiver  are  both  ground  to  a  plane.  G  is  the 
gaug*e  which  indicates  the  degree  of  exhaustion.  There  are  three 
valves,  the  first  at  the  bottom  of  the  barrel,  the  second  in  the 
piston,  and  the  third  at  the  top  of  the  barrel.  These  all  open  up- 
ward, allowing  the  air  to  pass  out,  but  preventing  its  return. 


169 


FIG.  176. 


256.  Operation. — When  the  piston  is  depressed,  the  air  below 
it,  by  its  increased  tension,  presses  down  the  first  valve,  and  opens 
the  second,  and  escapes  into  the  upper  part  of  the  barrel.  When 
the  piston  is  raised,  the  air  above  it  cannot  return,  but  is  pressed 
through  the  third  valve  into  the  open  air ;  while  the  air  in  the  re- 
ceiver and  pipe,  by  its  tension,  opens  the 
first  valve,  and  diffuses  itself  equally 
through  the  receiver  and  barrel.  An- 
other descent  and  ascent  only  repeat 
the  same  process;  and  thus,  by  a  suc- 
cession of  strokes,  the  air  is  nearly  all 
removed. 

The  exhaustion  can  be  made  more 
complete  if  the  first  and  second  valves 
are  opened  by  the  action  of  the  piston 
and  rod,  rather  than  by  the  tension  of 
the  air.  This  method  is  illustrated  by 
Fig.  176,  a  section  of  the  barrel  and  pis- 
ton. The  first  and  second  valves,  as 
shown  in  tKe  figure,  are  conical  or  pup- 
pet valves,  fitting  into  conical  sockets. 
The  first  has  a  long  stem  attached,  which 


170  PNEUMATICS. 

passes  through  the  piston  air-tight,  and  is  pulled  up  by  it  a  little 
way,  till  it  is  arrested  by  striking  the  top  of  the  barrel.  The  sec- 
ond valve  is  a  conical  frustum  on  the  end  of  the  piston-rod.  When 
the  rod  is  raised,  it  shuts  the  valve  before  moving  the  piston ; 
when  it  begins  to  descend,  it  opens  the  valve  again  before  giving 
motion  to  the  piston.  The  first  valve  is  shut  by  a  lever,  which 
the  piston  strikes  at  the  moment  of  its  reaching  the  top.  The  oil 
which  is  likely  to  be  pressed  through  the  third  valve  is  drained  off 
by  the  pipe  (on  the  right  in  both  figures)  into  a  cup  below  the 
pump. 

257.  Rate  of  Exhaustion. — The  quantity  removed,  by  suc- 
cessive strokes,  and  also  the  quantity  remaining  in  the  receiver, 
diminishes  in  the  same  geometrical  ratio.    For,  of  the  air  occupy- 
ing the  barrel  and  receiver,  a  barrel-full  is  removed  at  each  stroke, 
and  a  receiver-full  is  left.    If,  for  example,  the  receiver  is  three 
times  as  large  as  the  barrel,  the  air  occupies  four  parts  before  the 
descent  of  the  piston;  and  by  the  first  stroke  one-fourth  is  re- 
moved, and  three-fourths  are  left.    By  the  next  stroke,  three- 
fourths  as  much  will  be  removed  as  before  (|  of  f ,  instead  of  J  of 
the  whole),  and  so  on  continually.    The  quantity  left  obviously 
diminishes  also  in  the  same  ratio  of  three-fourths.    In  general,  if 
#  expresses  the  capacity  of  the  barrel,  and  r  that  of  the  receiver 

r 
and  connecting-pipe,  the  ratio  of  each  descending  series  is  •= . 

With  a  given  barrel,  the  rate  of  exhaustion  is  obviously  more 
rapid  as  the  receiver  is  smaller.  If  the  two  were  equal,  ten  strokes 
would  rarefy  the  air  more  than  a  thousand  times.  For  (|)10  = 

ToV?' 

As  a  term  of  this  series  can  never  reach  zero,  a  complete  ex- 
haustion can  never  be  effected  by  the  air-pump ;  but  in  the  best 
condition  of  a  well-made  pump,  it  is  not  easy  to  discover  by  the 
gauge  that  the  vacuum  is  not  perfect. 

258.  Experiments  with  the  Air-Pump. — By  the  air-pump 
a  great  variety  of  experiments  may  be  performed,  illustrative  of 
the  mechanical  properties  of  the  air.     The  obstruction  of  the  air 
being  removed,  light  and  heavy  bodies  are  seen  to  fall  with  equal 
rapidity ;  a  wheel  with  vanes  perpendicular  to  the  plane  of  rota- 
tion runs  as  freely  as  if  they  coincided  with  that  plane,  and  water 
boils  below  blood-heat.    The  weight  of  a  given  volume  of  air  is 
obtained  by  first  weighing  a  vessel  filled  with  air,  and  then  empty. 
The  pressure  of  air  in  every  direction  is  rendered  apparent  by 
many  striking  effects,  such  as  lifting  weights,  holding  together  the 
Magdeburg  hemispheres,  and  throwing  jets  of  water ;  also,  by  the 
difference  of  pressures  on  the  upper  and  lower  side,  bodies  are 


THE    AIR    CONDENSER. 


171 


FIG.  177. 


shown  to  weigh  less  in  air  than  in  a  vacuum.  And,  finally,  the 
tension  or  expansive  force  is  exhibited  by  experiments  equally  nu- 
merous and  interesting. 

259.  The  Air  Condenser. — While  the  air-pump  shows 
the  tendency  of  air  to  dilate  indefinitely,  as  the  com- 
pressing force  is  removed,  another  useful  instrument, 
the  condenser,  exhibits  the  indefinite  compressibility 
of  air.  Like  the  pump,  it  consists  of  a  barrel  and  piston, 
but  its  valves,  one  in  the  piston  and  one  at  the  bottom 
of  the  barrel,  open  downward.  Fig.  177  shows  the  exterior 
of  the  instrument.  If  it  be  screwed  upon  the  top  of  a 
strong  receiver  (Fig.  178),  with  a  stop-cock  connecting 
them,  air  may  be  forced  in,  and  then  secured  by  shutting 
the  stop-cock.  When  the  piston  is  depressed,  its  own 
valve  is  shut  by  the  increased  tension  of  the  air  beneath 
it,  and  the  lower  one  opened  by  the  same  force.  When 
the  piston  is  raised,  the  lower  valve  is  kept  shut  by  the 
condensed  air  in  the  receiver,  and  that  of  the  piston  is 
opened  by  the  weight  of  the  outer  air,  which  thus  gets 
admission  below  the  piston. 

The  quantity  of  air  in  the  receiver  increases  at  each 
stroke  in  an  arithmetical  ratio,  because  the  same  quan- 
tity, a  barrelrfull  of  common  air,  is  added  every  time  the  piston 
is  depressed.    A  small  Mariotte's  tube  is  attached  to  the  receiver, 
to  show  how  many  atmospheres  have  been  ad- 
mitted. FIG.  178. 

260.  Experiments   -with   the  Air  Con- 
denser.— If  the  receiver  be  partly  filled  with 
water,  and  a  pipe  from  the  stop-cock  extend  into 
it,  then  when  the  condenser  has  been  used  and 
removed,  and  the  stop-cock  opened,  a  jet  of  water 
will  be  thrown  to  a  height  corresponding  to  the 
tension  of  the  inclosed  air.     A  gas-bag  being 
placed  in  the  condenser,  then  filled  and  shut,  will 

become  flaccid  when  the  air  around  it  is  compressed.  A  thin  glass 
bottle,  sealed,  will  be  crushed  by  the  same  force.  By  these  and 
other  experiments  may  be  shown  the  effects  of  increased  tension. 

261.  Torricelli's  Experiment. — A  glass  tube  A  B  (Fig.  179) 
about  three  feet  long,  and  hermetically  sealed  at  one  end,  is  filled 
with  mercury,  and  then,  while  the  finger  is  held  tightly  on  the 
open  end,  it  is  inverted  in  a  cup  of  mercury.    On  removing  the 
finger  after  the  end  of  the  tube  is  beneath  the  surface  of  the  mer- 
cury, the  column  sinks  a  little  way  from  the  top,  and  there  re- 
mains.   Its  height  is  found  to  be  nearly  thirty  inches  above  the 


172 


PNEUMATICS. 


is  taken  to  expel 


FIG.  179. 


level  of  mercury  in  the  cup.  If  sufficient  care 
globules  of  air  from  the  liquid,  the 
space  above  the  column  in  the  tube  is 
as  perfect  a  vacuum  as  can  be  obtained. 
It  is  called  the  Torricellian  vacuum, 
from  Torricelli  of  Italy,  a  disciple  of 
Galileo,  who,  by  this  experiment,  dis- 
proved the  doctrine  that  nature  abhors 
a  vacuum,  and  fixed  the  limits  of  at- 
mospheric pressure. 

262.  Pressure  of  Air   Meas- 
ured.— The  column  is  sustained  in 
the  Torricellian  tube  by  the  pressure 
of  air  on  the  surface  of  mercury  in  the 
vessel ;  for  the  level  of  a  fluid  surface 
cannot  be  preserved  unless  there  is  an 
equal  pressure  on  every  part.     Hence, 
the  column  of  mercury  on  one  part, 
and  the  column  of  air  on  every  other 
equal  part,  must  press  equally.    To  de- 
termine, therefore,  the  pressure  of  air, 
we  have  only  to  weigh  the  column  of 
mercury,  and  measure  the  area  of  the 

mouth  of  the  tube.  If  this  is  carefully  done,  it  is  found  that  the 
weight  of  mercury  is  about  14.7  Ibs.  on  a  square  inch.  Therefore 
the  atmosphere  presses  on  the  earth  with  a  force  of  nearly  15 
pounds  to  every  square  inch,  or  more  than  2000  Ibs.  per  square 
foot. 

The  specific  gravity  of  mercury  is  about  13.6;  and  therefore 
the  height  of  a  column  of  water  in  a  Torricellian  tube  should  be 
13.6  times  greater  than  that  of  mercury,  that  is,  about  34  feet. 
Experiment  shows  this  to  be  true.  And  it  was  this  significant 
fact,  that  equal  weights  of  water  and  mercury  are  sustained  in 
these  circumstances,  which  led  Torricelli  to  attribute  the  effect  to 
a  common  force,  namely,  the  pressure  of  the  air. 

263.  Pascal's  Experiment. — As  soon  as  Torricelli's  discov- 
ery was  known,  Pascal  of  France  proposed  to  test  the  correct- 
ness of  his  conclusion,  by  carrying  the  apparatus  to  the  top  of  a 
mountain,  in  order  to  see  if  less  air  above  the  instrument  sustained 
the  mercury  at  a  less  height.    This  was  found  to  be  true;  the 
column  gradually  fell,  as  greater  heights  were  attained.     The  ex- 
periment of  Pascal  also  determined  the  relative  density  of  mercury 
and  air.     For  the  mercury  falls  one-tenth  of  an  inch  in  ascending 
87.2  feet;  therefore  the  weight  of  the  one-tenth  of  an  inch  of  mer- 


THE    BAROMETER.  173 

cury  was  balanced  by  the  weight  of  the  87.2  feet  of  air.  There- 
fore the  specific  gravities  of  mercury  and  air  (being  inversely  as 
the  heights  of  columns  in  equilibrium)  are  as  (87.2  x  12  x  10  =) 
10464  :  1.  In  the  same  way  it  is  ascertained  that  water  is  770 
times  as  dense  as  air.  These  results  can  of  course  be  confirmed 
by  directly  weighing  the  several  fluids,  which  could  not  be  done 
before  the  invention  of  the  air-pump. 

264.  The   Barometer. — When  the   Torricellian  tube  and 
basin  are  mounted  in  a  case,  and  furnished  with  a  graduated  scale, 
the  instrument  is  called  a  barometer.    The  scale  is  divided  into 
inches  and  tenths,  and  usually  extends  from  26  to  32  inches,  a 
space  more  than  sufficient  to  include  all  the  natural  variations  in 
the  weight  of  the  atmosphere.     By  attaching  a  vernier  to  the 
scale,  the  reading  may  be  carried  to  hundredths  and  thousandths 
of  an  inch,  as  is  commonly  done  in  meteorological  observations. 
By  observing  the  barometer  from  day  to  day,  and  from  hour  to 
hour,  it  is  found  that  the  atmospheric  pressure  is  constantly  fluc- 
tuating. 

As  the  meteorological  changes  of  the  barometer  are  all  com- 
prehended within  a  range  of  two  or  three  inches,  much  labor  has 
been  expended  in  devising  methods  for  magnifying  the  motions 
of  the  mercurial  column,  so  that  more  delicate  changes  of  atmo- 
spheric pressure  might  be  noted.  The  inclined  tube  and  the  wheel 
barometer  are  intended  for  this  purpose.  A  description  of  these 
contrivances,  however,  is  unnecessary,  as  they  are  all  found  to  be 
inferior  in  accuracy  to  the  simple  tube  and  basin. 

265.  Corrections  for  the  Barometer. — 

1.  For  change  of  level  in  the  basin. — The  numbers  on  the 
barometer  scale  are  measured  from  a  certain  zero  point,  which  is 
assumed  to  be  the  level  of  the  mercury  in  the  basin.    If  now  the 
column  falls,  it  raises  the  surface  in  the  basin ;  and  if  it  rises,  it 
lowers  it.    If  the  basin  is  broad,  the  change  of  level  is  small,  but 
it  always  requires  a  correction.     To  avoid  this  source  of  error,  the 
bottom  of  the  basin  is  made  of  flexible  leather,  with  a  screw  under- 
neath it,  by  which  the  mercury  may  be  raised  or  lowered,  till  its 
surface  touches  an  index  that  marks  the  zero  point.    This  adjust- 
ment should  always  be  made  before  reading  the  barometer. 

2.  For  capillarity. — In  a  glass  tube  mercury  is  depressed  by 
capillary  action  (Art.  227).    The  amount  of  depression  is  less  as 
the  tube  is  larger.     This  error  is  to  be  corrected  by  the  manufac- 
turer, the  scale  being  put  below  the  true  height  by  a  quantity 
equal  to  the  depression. 

There  is  a  slight  variation  in  this  capillary  error,  arising  from 


174  PNEUMATICS. 

the  fact  that  the  rounded  summit  of  the  column,  called  the  menis- 
cus, is  more  convex  when  ascending  than  when  descending.  To 
render  the  meniscus  constant  in  its  form,  the  barometer  should  be 
jarred  before  each  reading. 

3.  For  temperature. — As  mercury  is  expanded  by  heat  and  con- 
tracted by  cold,  a  given  atmospheric  pressure  will  raise  the  column 
too  high,  or  not  high  enough,  according  to  the  temperature  of  the 
mercury.    A  thermometer  is  therefore  attached  to  the  barometer, 
to  show  the  temperature  of  the  instrument.    By  a  table  of  correc- 
tions, each  reading  is  reduced  to  the  height  the  mercury  would 
have  if  its  temperature  was  32°  F. 

4.  For  altitude  of  station. — Before  comparing  the  observations 
of  different  places,  a  correction  must  be  made  for  altitude  of  sta- 
tion, because  the  column  is  shorter  according  as  the  place  is 
higher  above  the  sea  level. 

266.  The  Aneroid  Barometer. — This  is  a  small  and  port- 
able instrument,  in  appearance  a  little  like  a  large  chronometer. 
The  essential  part  of  this  barometer  is  a  flat  cylindrical  metallic 
box,  whose  upper  surface  is  corrugated,  so  as  to  be  yielding.    The 
box  being  partly  exhausted  of  air,  the  external  pressure  causes  the 
top  to  sink  in  to  a  certain  extent;  if  the  pressure  increases,  the 
surface  descends  a  little  more;  if  it  diminishes,  a  little  less.    These 
small  movements  are  communicated  by  a  system  of  levers  to  an 
index  on  the  graduated  face  of  the  barometer.    The  box  and  levers 
are  concealed  and  protected  within  the  outer  case.    As  might  be 
expected,  its  range  is  limited,  and  its  indications  not  perfectly  re- 
liable; but  for  obtaining  results  in  which  accuracy  is  not  essen- 
tial, its  lightness  and  convenient  form  and  size  recommend  it, 
especially  for  portable  uses. 

267.  Pressure  and  Latitude. — The  mean  pressure  of  the 
atmosphere  at  the  level  of  the  sea  is  very  nearly  30  inches.    But  ifc 
is  not  the  same  at  all  latitudes.    From  the  equator  either  north- 
ward or  southward,  the  mean  pressure  increases  to  about  latitude 
30°,  by  a  small  fraction  of  an  inch,  and  thence  decreases  to  about 
65°,  where  the  pressure  is  less  than  at  the  equator,  and  beyond 
that  it  slightly  increases.    This  distribution  of  pressures  in  zones 
is  due  to  the  great  atmospheric  currents,  caused  by  heat  in  con- 
nection with  the  earth's  rotation  on  its  axis. 

The  amount  of  variation  in  barometric  pressure  is  very  unequal 
in  different  latitudes ;  and  in  general,  the  higher  the  latitude,  the 
greater  the  variation.  Within  the  tropics  the  extreme  range 
scarcely  ever  exceeds  one-fourth  of  an  inch,  while  at  latitude  40° 
it  is  more  than  two  inches,  and  in  higher  latitudes  even  reaches 
three  inches. 


VARIATIONS    OF    THE    BAROMETER.  175 

268.  Diurnal  Variation. — If  a  long  series  of  barometric  ob- 
servations be  made,  and  the  mean  obtained  for  each  hour  of  the 
day,  the  changes  caused  by  weather  become  eliminated,  and  the 
diurnal  oscillation  reveals  itself.     It  is  found  that  the  pressure 
reaches  a  maximum  and  a  minimum  twice  in  24  hours.    The 
times  of  greatest  pressure  are  from  9  to  10,  and  of  least  pressure 
from  3  to  4,  both  A.  M.  and  p.  M.    In  tropical  climates  this  varia- 
tion is  very  regular,  though  small ;  but  in  the  temperate  zones  the 
irregular  fluctuations  of  weather  conceal  it  in  a  great  degree. 

This  double  oscillation  is  the  mingled  effect  of  heat  and 
moisture,  each  of  which  alone  would  produce  a  single  oscillation 
extending  through  the  entire  day. 

In  some  countries  of  the  torrid  zone  there  is  a  regular  annual 
oscillation  of  the  barometer;  but  in  the  temperate  zones  this  is 
scarcely  perceptible. 

269.  The  Barometer  and  the  Weather. — The  changes  in 
the  height  of  the  barometer  column  depend  directly  on  nothing 
else  than  the  atmospheric  pressure.    But  these  changes  of  pressure 
are  due  to  several  causes,  such  as  wind  and  changes  of  temperature 
and  moisture. 

The  practice  formerly  prevailed  of  engraving  at.  different  points 
of  the  barometer  scale  several  words  expressive  of  states  of  weather, 
"fair,  rain,  frost,  wind,"  &c.  But  such  indications  are  worthless, 
being  as  often  false  as  true ;  this  is  evident  from  the  fact  that  the 
height  of  the  column  would  be  changed  from  one  kind  of  weather 
to  another  by  simply  carrying  the  instrument  to  a  higher  or  lower 
station. 

No  general  system  of  rules  can  be  given  for  anticipating 
changes  of  weather  by  the  barometer,  which  would  be  applicable 
in  different  countries.  Eules  found  in  English  books  are  of  very 
little  value  in  America. 

Severe  and  extensive  storms  are  almost  always  accompanied  by 
a  fall  of  the  barometer  while  passing,  and  succeeded  by  a  rise  of 
the  barometer. 

270.  Heights  Measured  by  the  Barometer.— Since  mer- 
cury is  10464  times  as  heavy  as  air  (Art.  263),  if  the  barometer  is 
carried  up  until  the  mercury  falls  one  inch,  it  might  be  inferred 
that  the  ascent  is  10464  inches,  or  87-£  feet.    This  would  be  the 
case  if  the  density  were  the  same  at  all  altitudes.    But,  on  account 
of  diminished  pressure,  the  air  is  more  and  more  expanded  at 
greater  heights.    Besides  this,  the  height  due  to  a  given  fall  of  the 
mercury  varies  for  many  reasons,  such  as  the  temperature  of  the 
air,  the  temperature  of  the  mercury,  the  elevation  of  the  stations, 
and  their  latitude.    Hence,  the  measurement  of  heights  by  the 


176  PNEUMATICS. 

barometer  is  somewhat  troublesome,  and  not  always  to  be  relied 
on.  Formulae  and  tables  for  this  purpose  are  to  be  found  in  prac- 
tical works  on  physics. 

271.  The  Gauge  of  the  Air-Pump. — The  Torricellian  tube 
is  employed  in  different  ways  as  a  gauge  for  the  air-pump,  to  indi- 
cate the  degree  of  exhaustion.  In  Fig.  175  the  gauge  G  is  a  tube 
about  33  inches  long,  both  ends  of  which  are  open,  the  lower  im- 
mersed in  a  cup  of  mercury,  and  the  upper  communicating  with 
the  'interior  of  the  receiver.  As  the  exhaustion  proceeds,  the 
pressure  is  diminished  within  the  tube,  and  the  external  air  raises 
the  mercury  in  it.  A  perfect  vacuum  would  be  indicated  by  a 
height  of  mercury  equal  to  that  of  the  barometer  at  the  time. 

Another  kind  of  gauge  is  a  barometer  already  filled,  the  basin 
of  which  is  open  to  the  receiver.  As  the  tension  of  air  in  the  re- 
ceiver is  diminished,  the  column  descends,  and  would  stand  at  the 
same  level  in  both  tube  and  basin,  if  the  vacuum  were  perfect. 

A  modified  form  of  the  last,  called  the  siphon  gauge,  is  the 
best  for  measuring  the  rarity  of  the  air  in  the  receiver  when  the 
vacuum  is  nearly  perfect.  Its  construction  is  shown  by  Fig.  180. 
The  top  of  the  column,  A,  is  only  5  or  6  inches  above 
the  level  of  B  in  the  other  branch  of  the  recurved  tube. 
As  the  air  is  withdrawn  from  the  open  end  C,  the  ten- 
sion at  length  becomes  too  feeble  to  sustain  the  col- 
umn ;  it  then  begins  to  descend,  and  the  mercury  in 
the  two  branches  approaches  a  common  level. 

272.  Buoyant  Power  of  the  Air. — If  a  large 
and  a  small  body  are  in  equilibrium  on  the  two  arms 
of  a  balance,  and  the  whole  be  set  under  a  receiver, 
and  the  air  be  removed,  the  larger  body  will  prepon- 
derate, showing  that  .it  is  really  the  heaviest.  Their  apparent 
equality  of  weight  when  in  the  air  is  owing  to  its  buoyant  power ; 
for  air,  like  water  and  all  fluid  substances,  diminishes  the  apparent 
weight  of  an  immersed  body  by  just  the  weight  of  the  displaced 
fluid.  Hence,  the  larger  the  body,  the  more  weight  it  loses. 

It  follows  that  if  a  body  weighs  less  than  the  displaced  air,  it 
will  rise  just  as  light  bodies  do  in  water.  It  is  in  this  way  that 
balloons  are  made  to  ascend.  By  the  use  of  a  large  volume  of  hy- 
drogen, inclosed  in  a  silk  envelope,  rendered  air-tight  by  varnish, 
a  car  with  several  persons  in  it  can  be  carried  to  a  great  height. 
The  greatest  height  ever  attained  is  about  23000  feet,  or  nearly 
4.5  miles.  The  mercury  of  a  barometer  at  that  height  falls  to  12.5 
inches. 


BELLOWS    AND    SIPHON. 


177 


CHAPTER    II. 

INSTRUMENTS  WHOSE  OPERATION    DEPENDS    ON    THE    PROP- 
ERTIES  OF  AIR. 

273.  Pneumatic  Instruments. — Besides  the  apparatus  de- 
scribed in  the  foregoing  chapter,  by  the  aid  of  which  the  proper- 
ties of  the  air  are  discovered,  there  are  several  articles  in  common 
use  whose  utility  depends  more  or  less  on  the  same  properties,  and 
which  serve  as  good  illustrations  of  the  principles  already  pre- 
sented. 

274.  The  Bellows.— The  simple  or  hand-bellows  consists  of 
two  boards  or  lids  hinged  together,  and  having  a  flexible  leather 
round  the  edges,  and  a  tapering  tube  through  which  the  air  is 
driven  out.    In  the  lower  board  there  is  a  hole  with  a  valve  lying 
on  it,  which  can  open  inward.     On  separating  the  lids,  the  air  by 
its  pressure  instantly  lifts  the  valve  and  fills  the  space  between 
them;  but  when  they  are  pressed  together,  the  valve  shuts,  and 
the  air  is  compelled  to  escape  through  the  pipe.    The  stream  is 
intermittent,  passing  out  only  when  pressure  is  applied. 

The  compound  bellows,  used  for  forges  where  a  constant  stream 
is  needed,  are  made  with  two  compartments.  The  partition  C  T 
(Fig.  181)  is  fixed,  and 

has  in  it  a  valve   V  Fro.  181. 

opening  upward.  The 
lower  lid  has  also  a 
valve  V  opening  up- 
ward, and  the  upper 
one  is  loaded  with 
weights.  The  pipe  T 
is  connected  with  the 
upper  compartment. 
As  the  lower  lid  is 
raised  by  the  rod  A  B, 
which  is  worked  by  the 
lever  E  B,  the  air  in 
the  lower  part  is  crowded  through  V  into  the  upper  part,  whence 
it  is  by  the  weights  pressed  through  the  pipe  T7  in  a  constant 
stream.  When  the  lower  lid  falls,  the  air  enters  the  lower  com- 
partment by  the  valve  V. 

275.  The  Siphon.— If  a  bent  tube  ABC  (Fig.  182)  be  filled, 
and  one  end  immersed  in  a  vessel  of  water,  the  liquid  will  be  dis- 


178 


PNEUMATICS. 


FIG.  182. 


charged  through  the  tube  so  long  as  the  outer  end  is  lower  than 
the  level  in  the  vessel.  Such  a  tube  is  called  a  siplwn,  and  is 
much  used  for  removing  a  liquid  from  the  top  of  a  reservoir  with- 
out disturbing  the  lower  part.  The  height 
of  the  bend  B  above  the  fluid  level  must 
be  less  than  34  feet  for  water,  and  less 
than  30  inches  for  mercury.  The  reasons 
for  the  motion  of  the  water  are,  that  the 
atmosphere  is  able  to  sustain  a  column 
higher  than  E  B,  and  that  C  B  is  longer 
than  E  B.  The  two  pressures  on  the 
highest  cross-section  B  of  the  tube  are 
unequal.  For  the  atmospheric  pressure 
at  E  is  able  to  sustain  34  feet  of  water, 
and  therefore  at  B  exerts  a  pressure  equal 
to  34  -  E  B  toward  the  right.  At  C  the 
air  also  presses  upward  with  a  force  equal 
to  34  feet,  and  therefore  at  B  it  exerts  a 
pressure  to  the  left  equal  to  34  —  C  B. 
Subtracting  34  —  C  B  from  34  —  E  B, 
we  have  the  remainder,  C  D,  for  the  ex- 
cess of  pressure  to  the  right,  or  outward 
from  the  vessel.  Therefore  the  water  will 
flow  with  a  velocity  due  to  the  weight  of 
D  C\  hence,  the  velocity  diminishes  as 
the  vessel  empties. 

If  the  tube  is  small,  it  may  be  filled  by  suction,  after  the  end  A 
is  immersed.  If  it  is  large,  it  may  be  inverted  and  filled,  and  then 
secured  by  stop-cocks,  till  the  end  is  beneath  the  water. 

276.  Intermitting  Springs. — Springs  which  flow  freely  for  a 
time,  and  then  cease  for  a  certain  interval,  after  which  they  flow 
again,  are  found  in  some  cases  to  operate  on  the  principle  of  the 
siphon.    Suppose  a  reservoir  or  hollow  in  the  interior  of  a  hill, 
having  a  siphon-shaped  outlet.     It  is  obvious,  upon  hydrostatic 
principles,  that  no  water  will  be  discharged  until  the  fluid  has 
reached  a  level  in  the  reservoir  as  high  as  the  top  of  the  bend  in 
the  outlet.    Then  it  will  begin  to  run  out,  and  will  continue  to 
run  until  the  water  has  descended  to  the  level  of  the  outlet ;  after 
which  no  more  water  will  be  discharged  until  enough  has  collected 
to  reach  the  higher  level,  as  before. 

277.  The  Suction  Pump. — The  section  (Fig.  183)  exhibits 
the  construction  of  the  common  suction  pump.    By  means  of  a 
lever,  the  piston  P  is  moved  up  and  down  in  the  tube  A  V.     In 


SUCTION   AND   FORCING   PUMPS. 


179 


FIG.  183. 


A 


the  piston  is  a  valve  opening  upward,  and  at  the  top  of  the  pipe 

H  G  is  another  valve,  also  opening  upward.     The  latter  must  be 

at  a  less  height  than  34  feet  above  the  water  C.    When  the  piston 

is  raised,  its  valve  is  kept  shut  by  the  weight 

of  air  above,  and  the  atmospheric  pressure  at 

G  lifts  a  column  of  water  C  H  to  such  a  height 

that  its  weight,  added  to  the  tension  of  the 

rarefied  air,  HP,  equals  34  feet  of  water. 

When  P  descends,  the  air  below  is  prevented 

from  returning  by  the  lower  valve,  and  escapes 

through  the  piston.    The  piston  being  raised 

again,  the  water  rises  still  higher,  till  at  length 

it  passes  through  the  valve,  and  the  piston  dips 

into  it;  after  this  it  is  lifted  directly  to  the 

discharge  pipe  89  without  the  intervention  of 

the  air. 


278.    Calculation  of  the  Force.— Let 

the  whole  atmospheric  pressure  be  represented 
by  34  (its  equivalent  in  feet  of  water),  and  the 
height  of  water,  G  H,  by  li.  Since  the  tension 
of  air  in  the  tube,  added  to  Ji,  equals  34,  there- 
fore the  tension  =  34  —  A;  and  this  force  is 
exerted  upward  on  the  lower  side  of  the  piston ; 
while  the  downward  pressure  on  the  top  of  the 
piston  =  34.  The  difference  of  the  two  =  Ji, 
which  is,  therefore,  the  height  of  water,  whose 
downward  pressure  is  to  be  overcome.  We  ar- 
rive at  the  same  result  if  the  water  is  above  -jz.~n~Z.~z_~Z-T 
the  piston  at  A,  when  h  =  A  C.  For,  in  this 
case,  the  pressure  upward  on  P  =  34  —  P  (7;  while  the  down- 
ward pressure  =  34  +  A  P ;  and  the  difference  between  them  is 
P  C  +  A  P  —  Ji.  Therefore,  in  every  case,  the  force  required  to 
lift  the  piston  and  column  of  water  is  that  which  would  be  re- 
quired to  lift  the  same  weight  in  any  other  way.  The  atmosphere 
has  no  other  agency  than  to  furnish  a  convenient  mode  of  apply- 
ing the  force. 

If  d  =  the  diameter  of  the  piston,  in  decimals  of  a  foot,  then 
{-  TT  d*  =  its  area;  \  rr  d*  h  =  the  cubic  feet  of  water;  and 
J  TT  d2  li  x  62.5  =  Hie  pounds  of  water. 


279.  The  Forcing  Pump. — The  piston  of  the  forcing  pump 
(Fig.  184)  is  solid,  and  the  upper  valve  V  opens  into  the  side 
pipe  V  S.  In  the  ascent  of  the  piston,  the  water  is  raised  as  in 
the  suction  pump ;  but  in  its  descent,  a  force  must  be  applied 


180 


PNEUMATICS. 


to  press  the  water  which  is  above  F  into  the  side  pipe 
through  V. 

As  in  the  suction  pump,  the  force  expend-  FlG- 184- 

ed  is  that  required  to  lift  J  n  d?  li  x  62.5 
pounds  of  water.  But  the  two  differ  in  this 
respect :  in  the  suction  pump  the  force  is  all 
expended  in  raising  the  piston ;  in  the  forcing 
pump  the  force  is  divided,  and  the  column 
below  P  is  lifted  while  the  piston  ascends, 
and  that  above  P  while  it  descends. 

The  piston  is  only  one  of  many  contriv- 
ances for  producing  rarefaction  of  air  in  a 
pump-tube;  but  since  it  is  the  most  simple 
and  most  easily  kept  in  repair,  the  piston- 
pump  is  generally  preferred  to  any  other. 

280.  The  Fire-Engine.— This  machine 
generally  consists  of   one  or  more    forcing 
pumps,  with  a  regulating  air-vessel,  though 
the  arrangement  of  parts  is  exceedingly  varied. 
Fig.  185  will  illustrate  the  principles  of  its 
construction.    As  the  piston,  P,  ascends,  the 
water  is  raised  through  the  valve,  V,  by  at- 
mospheric pressure.    As  P  descends,  the  water 
is  driven  through  F  into  the  air-vessel,  M, 
whence  by  the  condensed  air  it  is  forced  out 
without  interruption  through  the  hose-pipe,  L. 
The  piston  Pf  operates  in  the  same  way  by 

alternate  movements.  The  piston-rods  are  attached  to  a  lever 
(not  represented),  to  which  the  strength  of  several  men  can  be  ap- 
plied at  once  by  means  of  hand-bars  called  brakes. 

The  air-vessel  may  be  attached 

to  any  kind  of  pump,  whenever  it  FlG- 185. 

is  desired  to  render  the  stream  con- 
stant. 

281.  Hero's  Fountain.— The 

condensation  in  the  air-vessel,  from 
which  water  is  discharged,  may  be 
produced  by  the  weight  of  a  column 
of  water.  An  illustration  is  seen  in 
Hero's  fountain,  Fig.  186.  A  ver- 
tical column  of  water  from  the  ves- 
sel, Ay  presses  into  the  air-vessel,  B, 
and  condenses  the  air  more  or  less, 
accprding  to  the  height  of  A  B. 


MANOMETERS. 


181 


From  the  top  of  this  vessel  an  air-tube  conveys  the  force  of  the 
compressed  air  to  a  second  air-vessel,  (7, 
which  is  nearly  full  of  water,  and  has  a  jet- 
pipe  rising  from  it.  Since  the  tension  of 
air  in  C  is  equal  to  that  in  B,  a  jet  will  be 
raised  which,  if  unobstructed,  would  be 
equal  in  height  to  the  compressing  column, 
AS. 

This  plan  has  been  employed  to  raise 
water  from  a  mine  in  Hungary,  and  hence 
called  "  the  Hungarian  machine." 

282.  Manometers. — These  are  instru- 
ments for  measuring  the  tension  of  gases 
or  vapors.    In  one  kind  of  manometer  the 
law  of  Mariotte  is  employed.    The  tube 
A  B  (Fig.  187),  closed  at  the  top,  has  its 
open  end  beneath  the  surface  of  mercury 
in  the  closed  cistern   C.    The  vessel  D, 
containing  the  gas  or  vapor  whose  tension 
is  to  be  measured,  communicates  with  the 
top  of  the  cistern.    If  the  mercury  is  at 
the  same  level  in  the  cistern  and  tube,  the 
pressure  equals  one  atmosphere.    As  the 
tension  in  D  increases,  the  column  in  A  B 
rises,  and  compresses  the  air  in  the  tube. 
The  tension  of  the  air  in  A  B  above  the 

mercury,  together  with  the  weight  of  the  mercury  above  the  level 
in  the  cistern,  is  equal  to  the  tension  in  D;  so  that  the  number 
(2)  will  not  be  in  the  middle  point  between  (1) 
and  the  top,  but  somewhat  below.     A  scale  of  FIG.  187. 

atmospheres  is  calculated  according  to  the  pro- 
portions of  the  instrument,  and  placed  by  the 
side  of  the  tube. 

283.  Apparatus  for  Preserving  a  Con- 
stant Level.— Let  A  B  (Fig.  188)  be  a  reser- 
voir which  supplies  a  liquid  to  the  vessel  C  D ; 
and  suppose  it  is  desired  to  preserve  the  level  at 
the  point  C  in  the  vessel,  while  the  liquid  is  dis- 
charged from  it  irregularly  or  at  intervals.    This 
is  accomplished  by  letting  the  discharge  pipe  E 
enter  C  D  below  the  required  level  C9  while  the 
air  is  supplied  to  the  reservoir  only  by  a  tube 
F  B,  which  just  reaches  that  level.    So  long  as 


183 


PNEUMATICS. 


the  liquid  in  C  D  is  below  (7,  it  is 
at  a  greater  depth  from  the  sur- 
face A  than  B  is,  and  therefore  the 
pressure  is  greater,  and  the  liquid 
will  run  from  E,  and  air  enter  at  B. 
But  when  the  vessel  is  filled  to  O, 
the  hydrostatic  pressures  at  C  and 
B  are  equal ;  it  is  therefore  impos- 
sible that  the  water  should  over- 
come the  air  at  G  and  pass  out,  and 
that  the  air  should  at  the  same  time 
overcome  the  water  at  B,  and  pass 
in.  Hence,  if  G  discharges  more 
slowly  than  E,  it  is  immaterial 
whether  water  is  running  from  G 
or  not;  the  vessel  will  remain  al- 
ways filled  to  the  level  G  B. 


FIG.  183. 


CHAPTER    III. 

THE  ATMOSPHERE.— ITS  QUANTITY,  HEIGHT,  AND  MOTIONS. 

284.  Quantity  of  the  Atmosphere. — Since  the  air  sus- 
tains a  column  of  mercury  thirty  inches  high,  the  weight  of  the 
whole  atmosphere  is  equal  to  that  of  a  stratum  of  mercury  thirty 
inches  thick  covering  the  globe.    The  thickness  is  relatively  so 
small  that  the  volume  of  the  stratum  may  be  reckoned  as  that  of 
a  parallelepiped,  thirty  inches  in  height,  and  having  a  base  equal 
to  the  surface  of  the  earth. 

Letting  R  =  the  radius  of  the  earth,  and  h  =  the  depth  of 
mercury,  the  earth's  surface  =  4  TT  J?2,  and  the  volume  of  mercury 
=  4  n  R9  Ji  =  4  x  3.14159  x  (3956  x  5280)2  x  2.5  cubic  feet. 

This  multiplied  by  62.5  x  13.6,  the  weight  of  a  cubic  foot  of 
mercury,  gives  about  11,650,000,000,000,000,000  Ibs.  This  is, 
therefore,  the  weight  of  the  earth's  atmosphere. 

285.  Virtual  Height  of  the  Atmosphere.— When  two 
fluid  columns  are  in  equilibrium  with  each  other,  their  heights 
are  inversely  as  their  specific  gravities  (Art.  221).    The  specific 
gravity  of  mercury  is  10464  times  that  of  the  air  at  the  ocean 
level.    Therefore,  if  the  air  had  the  same  density  in  all  parts,  its 
height  would  be  found  by  the  proportion, 

1  :  10464  :  :  2.5  :  26160  feet, 


HEIGHT    AND    DENSITY    OF    THE    AIR.          183 

which  is  almost  five  miles.  Hence,  the  quantity  of  the  entire  at- 
mosphere of  the  earth  is  pretty  correctly  conceived  of  when  we 
imagine  it  having  the  density  of  that  which  surrounds  us,  and 
reaching  to  the  height  of  five  miles. 

286.  Decrease  of  Density. — But  the  atmosphere  is  very 
far  from  being  throughout  of  uniform  density.  The  great  cause 
of  inequality  is  the  decreasing  weight  of  superincumbent  air  at 
increasing  altitudes.  The  law  of  diminution  of  density,  arising 
from  this  cause,  is  the  following: 

The  densities  of  the  air  decrease  in  a  geometrical  as  the  altitudes 
increase  in  an  arithmetical  ratio.  For,  let  us  suppose  the  air  to  be 
divided  into  horizontal  strata  of  equal  thickness,  and  so  thin  that 
the  density  of  each  may  be  considered  as  uniform  throughout. 
Let  a  be  the  weight  of  the  whole  column  from  the  top  to  the 
earth,  ~b  the  weight  of  the  whole  column  above  the  lowest  stratum, 
c  that  of  the  whole  column  above  the  second,  &c.  Then  the 
weight  of  the  lowest  stratum  is  a  —  b,  and  the  weight  of  the 
second  is  b  —  c,  &c.  Now  the  densities  of  these  strata,  and  there- 
fore their  weights  (since  they  are  of  equal  thickness),  are  as  the 
compressing  forces ;  or, 

a  —  b:b  — c::b:c; 
/.  a  c  —  b  c  =  b*  —  b  c ;  /.  a  c  =  V ; 

.'.a:b::b:c; 

in  the  same  way,  b :  c  •- :  c  :  d\ 

that  is,  the  weights  of  the  entire  columns,  from  the  successive 
strata  to  the  top  of  the  atmosphere,  form  a  geometrical  series; 
therefore,  the  densities  of  the  successive  strata,  varying  as  the  com- 
pressing forces,  also  form  a  geometrical  series.  If,  therefore,  at  a 
certain  distance  from  the  earth,  the  air  is  twice  as  rare  as  at  the 
surface  of  the  earth,  at  twice  that  distance  it  will  be  four  times  as 
rare,  at  three  times  that  distance  eight  times  as  rare,  &c. 

By  barometric  observations  at  different  altitudes,  it  is  found 
that  at  the  height  of  three  and  a  half  miles  above  the  earth  the  air 
is  one-half  as  dense  as  it  is  at  the  surface.  Hence,  making  an 
arithmetical  series,  with  34  for  the  common  difference,  to  denote 
heights,  and  a  geometrical  series,  with  the  ratio  of  J,  to  denote 
densities,  we  have  the  following : 

Heights,     3A,  7,  10£,  14,  m,  21,  24J,  28,  31J-,    35. 
Densities,    £,  \,    |,    T'B,   fa  fa  TJ5,  TJff,  ^,  TTJ'3?. 

According  to  this  law,  the  air,  at  the  height  of  35  miles,  is  at 
least  a  thousand  times  less  dense  than  at  the  surface  of  the  earth. 
It  has,  therefore,  a  thousand  times  less  weight  resting  upon  it ;  in 
other  words,  only  one-thousandth  part  of  the  air  exists  above  that 
height 


184  PNEUMATICS. 

2B7.  Actual  Height  of  the  Atmosphere.— The  foregoing 
law,  founded  on  that  of  Mariotte,  cannot,  however,  be  applicable 
except  to  moderate  distances.  If  it  were  strictly  true,  the  atmo- 
sphere would  be  unlimited.  But  that  is  impossible  on  a  revolving 
body,  since  the  centrifugal  force  must  at  some  distance  or  other 
equal  the  force  of  gravity,  and  thus  set  a  limit  to  the  atmosphere ; 
and  that  limit  in  the  case  of  the  earth  is  more  than  20,000  miles 
high.  The  actual  height  of  the  atmosphere  is  doubtless  far  below 
this ;  for  there  can  be  none  above  the  point  where  the  repellency 
of  the  particles  is  less  than  their  weight ;  and  the  repellency  di- 
minishes just  as  fast  as  the  density,  while  the  weight  diminishes 
very  slowly.  The  highest  portions  concerned  in  reflecting  the 
sunlight  are  about  45  miles  above  the  earth.  But  there  is  reason 
to  believe  that  the  air  extends  much  above  that  height,  probably 
100  or  200  miles  from  the  earth. 

288.  The  Motions  of  the  Air. — The  air  is  never  at  rest. 
When  in  motion,  it  is  called  wind.    The  equilibrium  of  the  atmo- 
sphere is  disturbed  by  the  unequal  heat  on  different  parts  of  the 
earth.    The  air  over  the  hotter  portions  becomes  lighter,  and  is 
therefore  pressed  upward  by  the  cooler  and  heavier  air  of  the  less 
heated  regions.    And  the  motions  thus  caused  are  modified  as  to 
direction  and  velocity  by  the  rotation  of  the  earth  on  its  axis. 

289.  The  Trade  Winds. — The  most  extensive  and  regular 
system  of  winds  on  the  earth  is  known  by  the  name  of  the  trade 
winds,  so  called  on  account  of  their  great  advantage  to  commerce. 
They  are  confined  to  a  belt  about  equal  in  width  to  the  torrid 
zone,  but  whose  limits  are  four  or  five  degrees  further  north  than 
the  tropics. 

In  the  northern  half  of  this  trade-wind  zone  the  wind  blows 
continually  from  the  northeast,  and  in  the  southern  half  from  the 
southeast.  As  these  currents  approach  each  other,  they  gradually 
become  more  nearly  parallel  to  the  equator,  while  between  them 
there  is  a  narrow  belt  of  calms,  irregular  winds,  and  abundant 
rains. 

The  oblique  directions  of  the  trade  winds  are  the  combined 
effects  of  the  heat  of  the  torrid  zone  and  the  rotation  of  the  earth. 
The  cold  air  of  the  northern  hemisphere  tends  to  flow  directly 
south,  and  crowd  up  the  hot  air  over  the  equator.  In  like  manner, 
the  cold  air  of  the  southern  hemisphere  tends  to  flow  directly 
northward.  So  that  if  the  earth  were  at  rest,  there  would  be  north 
winds  on  the  north  side  of  the  equator,  and  south  winds  on  the 
south  side.  But  the  earth  revolves  on  its  axis  from  west  to  east, 
and  the  air,  as  it  moves  from  a  higher  latitude  to  a  lower,  has  only 
so  much  eastward  motion  as  the  parallel  from  which  it  came. 


THE    RETURN    CURRENTS.  185 

Therefore,  since  it  really  has  a  less  motion  from  the  west  than 
those  regions  over  which  it  arrives,  it  has  relatively  a  motion  from 
the  east.  This  motion  from  the  east,  compounded  with  the  motion 
from  the  north  on  the  north  side  of  the  equator,  and  with  that 
from  the  south  on  the  south  side,  constitutes  the  northeast  and 
southeast  tradewinds. 

The  limits  of  this  system  move  a  few  degrees  to  the  north  dur- 
ing the  northern  summer,  and  to  the  south  during  the  northern 
winter,  but  very  much  less  than  might  be  expected  from  the 
changes  in  the  sun's  declination. 

In  certain  localities  within  the  tropics  the  wind,  owing  to 
peculiar  configurations  of  coast  and  elevations  of  the  interior, 
changes  its  direction  periodically,  blowing  six  months  from  one 
point,  and  six  months  from  a  point  nearly  opposite.  The  monsoons 
of  southern  India  are  the  most  remarkable  example. 

290.  The  Return  Currents.— The  air  which  is  pressed  up- 
ward over  the  torrid  zone  must  necessarily  flow  away  northward 
and  southward  towards  the  higher  latitudes,  to  restore  the  equi- 
librium.    Hence,  there  are  south  winds  in  the  upper  air  on  the 
north  side  of  the  equator,  and  north  winds  on  the  south  side.    But 
these  upper  currents  are  also  oblique  to  the  meridians,  because, 
having  the  easterly  motion  of  the  equator,  they  move  faster  than 
the  parallels  over  which  they  successively  arrive,  so  that  a  motion 
from  the  west  is  combined  with  the  others,  causing  southwest 
winds  in  the  northern  hemisphere,  and  northwest  in  the  southern. 
These  motions  of  the  upper  air  are  discovered  by  observations 
made  on  high  mountains,  and  in  balloons,  and  by  noticing  the 
highest  strata  of  clouds.    It  is  to  be  borne  in  mind  that  although 
the  atmosphere  is  more  than  100  miles  high,  yet  the  lower  half 
does  not  extend  beyond  three  and  a  half  miles  above  the  earth 
(Art.  286). 

291.  Circulation  Beyond  the  Trade  Winds. — The  upper 
part  of  the  air  which  flows  away  from  the  equator  cannot  wholly 
retain  its  altitude,  because  of  the  diminishing  space  on  the  suc- 
cessive parallels.    About  latitude  30°,  it  is  so  much  accumulated 
that  it  causes  a  sensible  increase  of  pressure  (Art.  267),  and  begins 
to  descend  to  the  earth.    It  is  probable  that  some  of  the  descend- 
ing air  still  retains  its  oblique  motion  towards  higher  latitudes 
(for  the  prevailing  winds  of  the  northern  temperate  zone  are  from 
the  southwest,  and  of  the  southern  temperate  zone  from  the  north- 
west), while  a  part  joins  with  the  lower  air  which  is  moving 
towards  the  equator.     Only  so  much  of  the  rising  equatorial  mass 
can  flow  back  to  the  polar  regions  as  is  needed  to  supply  the 
comparatively  small  area  within  them.     On  account  of  the  sue- 


186  PNEUMATICS. 

cessive  descent  of  the  air  returning  from  the  equator,  there  is 
much  less  distinctness  and  regularity  in  the  general  circulation 
outside  of  the  torrid  zone  than  within  it.  Besides  this,  various 
local  causes,  such  as  mountain  ranges,  sea-coasts,  and  ocean  cur- 
rents, clear  and  cloudy  skies,  &c.,  mingle  their  effects  with  the 
more  general  circulation,  and  modify  it  in  every  possible  way. 

292.  Land  and  Sea  Breezes. — These  are  limited  circula- 
tions over  adjoining  portions  of  land  and  water,  the  wind  blowing 
from  the  water  to  the  land  in  the  day  time,  and  in  the  contrary 
direction  by  night.    When  the  sun  begins  to  shine  each  day,  it 
heats  the  land  more  rapidly  than  the  water.    Hence  the  air  on 
the  land  becomes  warmer  and  lighter  than  that  on  the  water,  and 
the  surface  current  sets  toward  the  land.    By  night  the  flow  is  re- 
versed, because  the  land  cools  most  rapidly,  and  the  air  above  it 
becomes  heavier  than  that  over  the  water.    These  effects  are  more 
striking  and  more  regular  in  tropical  countries,  but  are  common 
in  nearly  all  latitudes. 

293.  A  Current  Through  a   Medium. — There  are   some 
phenomena  relating  to  currents  moving  through  a  fluid,  either  of 
the  same  or  a  different  kind,  which  belong  alike  to  hydraulics  and 
pneumatics ;  a  brief  account  of  these  is  presented  here. 

If  a  stream  is  driven  through  a  medium,  it  carries  along  the 
adjoining  particles  by  friction  or  adhesion.  The  experiment  of 
Venturi  illustrates  this  kind  of  action,  as  it  takes  place  between 
the  particles  of  water.  A  reservoir  filled  with  water  has  in  it  an 
inclined  plane  of  gentle  ascent,  whose  summit  just  reaches  the 
edge  of  the  reservoir.  A  stream  of  water  is  driven  up  this  plane 
with  force  sufficient  to  carry  it  over  the  top ;  but  in  doing  so,  it 
takes  out  continually  some  part  of  the  water  of  the  reservoir,  and 
will  in  time  empty  it  to  the  level  of  the  lowest  part  of  the  stream. 
A  stream  of  air  through  air  produces  the  same  effect,  as  may  be 
shown  by  the  flame  of  a  lamp  near  the  stream  always  bending  to- 
ward it.  In  like  manner,  water  through  air  carries  air  with  it ; 
when  a  stream  of  water  is  poured  into  a  vessel  of  water,  air  is  car- 
ried down  in  bubbles ;  and  cataracts  carry  down  much  air,  which 
as  it  rises  forms  a  mass  of  foam  on  the  surface.  The  strong  wind 
from  behind  a  high  waterfall  is  owing  to  the  condensation  of  air 
brought  down  by  the  back  side  of  the  sheet. 

294.  Ventilators. — If  the  stream  passes  across  the  end  of  an 
open  tube,  the  air  within  the  tube  will  be  taken  along  with  the 
stream,  and  thus  a  partial  vacuum  formed,  and  a  current  estab- 
lished.   It  is  thus  that  the  wind  across  the  top  of  a  chimney  in- 
creases the  draught  within.    To  render  this  effect  more  uniformly 


CURRENTS    MEETING    A    SURFACE.  187 

successful,  by  preventing  the  wind  from  striking  the  interior  edge 
of  the  flue,  appendages,  called  ventilators, 
are  attached  to  the  chimney  top.    A  sim-  FIG.  189. 

pie  one,  which  is  generally  effectual,  con- 
sists of  a  conical  frustum  surrounding  the 
flue,  as  in  Fig.  189,  so  that  the  wind,  on 
striking  the  oblique  surface,  is  thrown 
over  the  top  in  a  curve,  which  is  convex 
upward.  The  same  mechanical  contriv- 
ance is  much  used  for  the  ventilation  of 
public  halls  and  the  holds  of  ships.  A 
horizontal  cover  may  be  supported  by  rods, 
at  the  height  of  a  few  inches,  to  prevent 
the  rain  from  entering. 

295.  A  Stream  Meeting  a  Surface. — Though  the  moving 
fluid  may  be  elastic,  yet,  when  it  meets  a  surface,  it  tends  to  follow 
it,  rather  than  to  rebound  from  it.     This  effect  is  partly  due  to 
adhesion,  and  partly  to  the  resistance  of  the  medium  in  which  the 
stream  moves.    It  will  not  only  follow  a  plane  or  concave  surface, 
but  even  one  which  is  convex,  provided  the  velocity  of  the  current 
is  not  too  great,  or  the  curvature  too  rapid.    A  stream  of  air, 
blown  from  a  pipe  upon  a  plane  surface,  will  extinguish  the  flame 
of  a  lamp  held  in  the  direction  of  the  surface  beyond  its  edge, 
while,  if  the  lamp  be  held  elsewhere  near  the  stream,  the  flame 
will  point  toward  the  stream,  according  to  Art.  293.     Hence,  snow 
is  blown  away  from  the  windward  side  of  a  tight  fence,  and  from 
around  trees. 

296.  Diminution  of  Pressure   on   a  Surface. — When  a 
stream  is  thus  moving  along  a  surface,  the  fluid  pressure  on  that 
surface  is  slightly  diminished.     This  is  proved  by  many  experi- 
ments.    If  a  curved  vane  be  suspended  on  a  pivot,  and  a  stream 
of  air  be  directed  tangentially  along  the  surface,  it  will  move  to- 
ward the  stream,  and  may  be  made  to  revolve  rapidly  by  repeating 
the  blast  at  each  half  revolution.    What  is  frequently  called  the 
pneumatic  paradox  is  a  phenomenon  of  the  same  kind.    A  stream 
of  air  is  blown  through  the  centre  of  a  disk,  against  another  light 
disk,  which,  instead  of  being  blown  off,  is  forcibly  held  near  to  it 
by  the  means.    The  pressure  is  diminished  by  all  the  radial  streams 
along  the  surface  contiguous  to  the  other  disk,  and  the  full  press- 
ure on  the  outside  preponderates.    Another  form  of  the  experi- 
ment is  to  blow  a  stream  of  air  through  the  bottom  of  a  hemi- 
spherical cup,  in  which  a  light  sphere  is  lying  loosely.    The  sphere 
cannot  be  blown  out,  but,  on  the  contrary,  is  held  in,  as  may  be 
seen  by  inverting  the  cup,  while  the  blast  continues.    It  appears 


188  PNEUMATICS. 

to  be  for  a  reason  of  the  same  sort  that  a  ball  or  a  ring  is  sustained 
by  a  jet  of  water.  It  lies  not  on  the  top,  but  on  the  side  of  the 
jet,  which  diminishes  the  pressure  on  that  side  of  the  ball,  so  that 
the  air  on  the  outside  keeps  it  in  contact.  The  tangential  force 
of  the  jet  causes  the  body  to  revolve  with  rapidity.  A  ball  can  be 
sustained  a  few  inches  high  by  a  stream  of  air. 

297.  Vortices  where  the  Surface   Ends. — As  a  current 
reaches  the  termination  of  the  surface  along  which  it  was  flowing, 
a  vortex  or  whirl  is  likely  to  occur  in  the  surrounding  medium 
behind  the  edge  of  the  surface.    Vortices  are  formed  on  water, 
whose  flow  is  obstructed  by  rocks ;  and  often  when  the  obstruct- 
ing body  is  at  a  distance  below  the  surface,  the  whirl  which  is  es- 
tablished there  is  communicated  to  the  top,  so  that  the  vortex  is 
seen,  while  its  cause  is  out  of  sight.     There  is  a  depression  at  the 
centre,  caused  by  the  centrifugal  force ;   and  if  the  rotation  is 
rapid,  a  spiral  tube  is  formed,  in  which  the  air  descends  to  great 
depths.    These  are  called  whirlpools.    In  a  similar  manner  whirls 
are  produced  in  the  air,  when  it  pours  off  from  a  surface.     The 
eddying  leaves  on  the  leeward  side  of  a  building  in  a  windy  day 
often  indicate  such  a  movement,  though  it  may  have  no  perma- 
nency, the  vortex  being  repeatedly  broken  up  and  reproduced. 

298.  Vortices  by  Currents  Meeting. — But  vortices  are 

also  formed  by  counteracting  currents  in  an  open  medium.  When 
an  aperture  is  made  in  the  middle  of  the  bottom  of  a  vessel,  as  the 
water  runs  toward  it,  the  filaments  encounter  each  other,  and 
usually,  though  not  invariably,  they  establish  a  rotary  motion, 
and  form  a  whirlpool.  Vortices  are  a  frequent  phenomenon  of 
the  atmosphere,  sometimes  only  a  few  feet  in  diameter,  in  other 
instances  some  rods  or  even  miles  in  width.  The  smaller  ones, 
occurring  over  land,  are  called  whirlwinds;  over  water,  ivater- 
spouts.  They  probably  originate  in  currents  which  do  not  exactly 
oppose  each  other,  but  act  as  a  couple  of  forces,  tending  to  produce 
rotation  (Art.  54). 

The  burning  of  a  forest  sometimes  occasions  whirlwinds,  which 
are  borne  away  by  the  wind,  and  maintain  their  rotation  for  miles. 
As  the  pressure  in  the  centre  is  diminished  by  the  centrifugal 
force,  substances  heavier  than  air,  as  leaves  and  spray,  are  likely  to 
be  driven  up  in  the  axis,  and  floating  substances,  as  cloud,  will  for 
the  same  reason  descend.  The  rising  spray  and  the  descending 
cloud  frequently  mark  the  progress  of  a  vortex  in  the  air,  as  it 
moves  over  a  lake  or  the  ocean.  Such  a  phenomenon  is  called  a 
water-spout. 

For  a  notice  of  cyclones,  see  Part  VIII,  on  Heat. 


PART  IV. 

SOUND 


CHAPTER  I. 

NATURE  AND  PROPAGATION  OF  SOUND. 

299.  Sound. — Vibrations. — The  impression  which  the  mind 
receives  through  the  organ  of  hearing  is  called  sound.  But  the 
same  word  is  constantly  used  to  signify  that  progressive  vibratory 
movement  in  a  medium  by  which  the  impression  is  produced,  as 
when  we  speak  of  the  velocity  of  sound. 

This  is  one  of  the  several  modes  of  motion  mentioned  in  Art.  4. 
The  vibrations  constituting  sound  are  comparatively  slow,  and  are 
often  perceived  by  sight  and  by  feeling  as  well  as  by  hearing.  For 
these  reasons,  the  true  nature  of  sound  is  investigated  with  far 
greater  ease  than  that  of  light,  electricity,  &c.  It  is  not  difficult 
to  discover  that  vibrations  in  the  medium  about  us  are  essential  to 
hearing ;  and  these  vibrations  are  always  traceable  to  the  body  in 
which  the  sound  originates.  A  body  becomes  a  source  of  sound 
by  producing  an  impulse  or  a  series  of  impulses  on  the  surrounding 
medium,  and  thus  throwing  the  medium  itself  into  motion.  A 
single  sudden  impulse  causes  &  noise,  with  very  little  continuance; 
an  irregular  and  rapid  succession  of  impulses,  a  crash,  or  roar,  or 
continued  noise  of  some  kind ;  but  if  the  impulses  are  rapid  and 
perfectly  equidistant,  the  effect  is  a  musical  sound.  In  most  cases 
of  the  last  kind  the  impulses  are  vibrations  of  the  body  itself;  and 
whatever  affects  these  vibrations  is  found  to  affect  the  sound  em- 
anating from  it ;  and  if  they  are  destroyed,  the  sound  ceases. 

If  we  rub  a  moistened  finger  along  the  edge  of  a  tumbler 
nearly  full  of  water,  or  draw  a  bow  across  the  strings  of  a  viol,  we 
can  procure  sounds  which  remain  undiminished  in  intensity  as 
long  as  the  operation  by  which  they  are  excited  is  continued.  In 
both  cases  the  vibrations  are  visible ;  those  of  the  tumbler  are 
plainly  seen  as  crispations  on  the  water  to  which  they  are  commu- 
nicated ;  the  string  appears  as  a  broad  shadowy  surface.  If  a  wire 
or  light  piece  of  metal  rests  against  a  bell  or  glass  receiver,  when 


190  SOUND. 

ringing,  it  will  be  made  to  rattle.  If  sand  be  strewed  on  a  hori- 
zontal plate  while  a  bow  is  drawn  across  its  edge,  the  sand  will  be 
agitated,  and  dance  over  the  surface,  till  it  finds  certain  places 
where  vibrations  do  not  exist.  Near  an  organ-pipe  the  tremor  of 
the  air  is  perceptible,  and  pipes  of  the  largest  size  jar  the  seats  and 
walls  of  an  edifice.  Every  species  of  sound  may  be  traced  to  im- 
pulsea  or  vibrations  in  the  sounding  body. 

300.  Sonorous  Bodies. — Two  qualities  in  a  body  are  neces- 
sary, in  order  that  it  may  be  sonorous.     It  must  have  a  form 
favorable  for  vibratory  movements,  and  sufficient  strength  of  elas- 
ticity. 

The  favorable  forms  are  in  general  rods  and  plates,  rather  than 
very  compact  masses,  like  spheres  and  cubes  ;  because  the  particles 
of  the  former  are  more  free  to  receive  lateral  movements  than 
those  of  the  latter,  which  are  constrained  on  every  side.  But  even 
a  thin  lamina  may  have  a  form  which  allows  too  little  freedom  of 
motion,  such  as  a  spherical  shell,  in  which  the  parts  mutually  sup- 
port each  other.  If  the  shell  be  divided,  the  hemispheres  are 
bell-shaped  and  very  sonorous. 

The  elasticity  of  some  materials  is  too  imperfect  for  continued 
vibration  ;  thus  lead,  in  whatever  form,  has  no  sonorous  quality. 
In  other  cases,  where  the  elasticity  is  nearly  perfect,  yet  it  is  a 
feeble  force,  and  hence  the  vibrations  are  slow  and  inaudible. 
Thus  india-rubber  is  quite  elastic,  but  its  force  is  feeble,  and  occa- 
sions but  little  sound. 

301.  Air  as  a  Medium  of  Sound. — There  must  not  only 
be  a  vibrating  body,  as  a  source  of  sound,  but  a  medium  for  its 
communication  to  the  organ  of  hearing.     The  ordinary  medium  is 
air.     Let  a  bell  mounted  with  a  hammer  and  mainspring,  so  as  to 
continue  ringing  for  several  minutes,  be  placed  on  a  thick  cushion 
under  the  receiver  of  an  air-pump.     The  cushion,  made  of  several 
thicknesses  of  woolen  cloth,  is  necessary  to  prevent  communica- 
tion through  the  metallic  parts  of  the  instrument.    As  the  pro- 
cess of  exhaustion  goes  on,  the  sound  of  the  bell  grows  fainter, 
and  at  length  ceases  entirely.    From  this  experiment  we  learn  that 
sound  cannot  be  propagated  through  a  vacant  space,  even  though 
'it  be  only  an  inch  or  two  in  extent ;  and  also  that  air  conveys 
sound  more  feebly  as  it  is  more  rare.    The  latter  is  proved  by  the 
faintness  of  sounds  on  the  tops  of  high  mountains.    Travelers 
among  the  Alps  often  observe  that  at  great  elevations  a  gun  can 
be  heard  only  a  small  distance.    The  fact  that  meteoric  bodies  are 
sometimes  heard  when  passing  over  at  the  height  of  40  or  50  miles 
does  not  conflict  with  the  above  statements ;  for  the  velocity  of 
meteors  is  vastly  greater  than  any  other  velocities  which  occur 


VELOCITY    OF    SOUND.  191 

within  the  earth's  atmosphere.  On  the  other  hand,  when  air  has 
more  than  the  natural  density,  it  conveys  sound  with  more  inten- 
sity, and  therefore  to  a  greater  distance.  In  a  diving-bell  sunk  to 
a  considerable  depth  a  whisper  is  painfully  loud. 

302.  Velocity  of  Sound  in  Air. — Sound  occupies  an  ap- 
preciable time  in  passing  through  air.     This  is  a  fact  of  common 
observation.     The  flash  of  a  distant  gun  is  seen  before  the  report 
is  heard.     Thunder  usually  follows  lightning  after  an  interval  of 
many  seconds;  but  if  the  electric   discharge  is  quite  near,  the 
lightning  and  thunder  are  almost  simultaneous.    If  a  person  is 
hammering  at  a  distance,  the  perceptions  of  the  blows  received  by 
the  eye  and  the  ear  do  not  generally  agree  with  each  other :  or  if 
in  any  case  they  do  agree,  it  will  be  observed  that  the  first  stroke 
seen  is  inaudible,  and  the  last  one  heard  is  invisible ;  for  it  re- 
quires just  the  time  between  two  strokes  for  the  sound  of  each  to 
reach  us.     Many  careful  experiments  were  made  in  the  18th  cen- 
tury to  determine  the  velocity  of  sound ;  but  as  the  temperature 
was  not  recorded,  they  have  but  little  value.     During  the  present 
century,  the  velocity  has  been  determined  by  several  series  of  ob- 
servations in  different  countries,  and  all  reduced  for  temperature 
to  the  freezing-point.     The  agreement  between  them  is  very  close, 
and  the  mean  of  all  is  1090  feet  per  second  at  32°  F. 

303.  Velocity  as  Affected  by  the  Condition  of  the  Air 
and  the  Quality  of  the  Sound. — 

Temperature  affects  the  velocity  of  sound;  the  latter  is  in- 
creased about  one  foot  (0.96  ft.)  for  each  degree  of  rise  in  the  tem- 
perature. Therefore,  in  most  New  England  climates,  the  velocity 
of  sound  varies  more  than  100  feet  during  the  year  on  this  ac- 
count. Probably  the  celebrated  experiments  of  Derham,  in  Lon- 
don, 1708,  who  made  the  velocity  1142  feet,  were  performed  in  the 
heat  of  summer. 

Wind  of  course  affects  the  velocity  of  sound  by  the  addition  or 
subtraction  of  its  own  velocity,  estimated  in  the  same  direction, 
because  it  transfers  the  medium  itself  in  which  the  sound  is  con- 
veyed. This  modification,  however,  is  only  slight,  for  sound 
moves  ten  times  faster  than  wind  in  the  most  violent  hurricane. 

But  other  changes  in  the  condition  of  the  air  produce  little  or 
no  effect.  Neither  pressure,  nor  moisture,  nor  any  change  of 
weather,  alters  the  velocity  of  sound,  though  they  may  affect  its 
intensity,  and  therefore  the  distance  at  which  it  can  be  heard. 
Falling  snow  and  rain*  obstruct  sound,  but  do  not  retard  it. 

All  kinds  of  sound — the  firing  of  a  gun — the  blow  of  a  ham- 
mer— the  notes  of  a  musical  instrument,  or  of  the  voice,  however 
high  or  low,  loud  or  soft,"  are  conveyed  at  the  same  rate.  That 


192  SOUND. 

sounds  of  different  pitch  are  conveyed  with  the  same  velocity 
was  conclusively  proved  by  Biot,  in  Paris,  who  caused  several  airs 
to  be  played  on  a  flute  at  one  end  of  a  pipe  more  than  3000  feet 
long,  and  heard  the  same  at  the  other  end  distinctly,  and  without 
the  slightest  displacement  in  the  order  of  notes,  or  intervals  of 
silence  between  them. 

304.  The  Calculated  Velocity. — For  several  years  there 
was  a  large  unexplained  difference  between  the  calculated  velocity 
of  sound  and  the  actual  velocity  as  determined  by  experiment. 
"While  the  latter  is,  as  already  stated,  1090  feet  per  second  at  the 
freezing-point,  calculation  gave  916  feet.     The  difference  was  at 
length  explained  by  La  Place,  who  ascertained  that  it  arises  from 
the  heat  developed  in  the  air  by  the  compression  which  it  under- 
goes.   The  calculations  previously  made  regarded  the  elasticity  as 
varying  with  the  density  alone,  according  to  Mariotte's  law,  as- 
suming that  the  temperature  remained  unchanged.     But  it  is  a 
wejl-known  fact  that  when  air  is  compressed,  a  part  of  its  latent 
heat  becomes  sensible,  and  raises  its  temperature.     If  the  conden- 
sation is  gradual,  the  heat  is  radiated  or  conducted  off,  especially 
if  in  contact  with  other  bodies ;  but  the  heat  developed  in  the 
propagation  of  sound  has  little  opportunity  to  escape,  and,  though 
without  continuance,  it  augments  the  elasticity  of  the  air,  so  as  to 
add  174  feet  to  the  velocity  of  sound  in  it. 

305.  Diffusion  of  Sound. — Sound  produced  in  the  open  air 
tends  to  spread  equally  in  all  directions,  and  will  do  so  whenever 
the  original  impulses  are  alike  on  every  side.    But  this  is  rarely 
the  case.     In  firing  a  gun,  the  first  impulse  is  given  in  one  direc- 
tion, and  the  sound  will  have  more  intensity,  and  be  heard  further 
in  that  direction  than  in  others.    It  is  ascertained  by  experiment, 
that  a  person  speaking  in  the  open  air  can  be  equally  well  heard 
at  the  distance  of  100  feet  directly  before  him,  75  feet  on  the  right 
and  left,  and  30  feet  behind  him ;  and  therefore  an  audience,  in 
order  to  hear  to  the  best  advantage,  should  be  arranged  within 
limits  having  these  proportions.    But,  as  will  be  seen  hereafter, 
this  rule  is  not  applicable  to  the  interior  of  a  building. 

Sound  is  also  heard  in  certain  directions  with  more  intensity, 
and  therefore  to  a  greater  distance,  if  an  obstacle  prevents  its  dif- 
fusion in  other  directions.  On  one  side  of  an  extended  wall  sound 
is  heard  further  than  if  it  spread  on  both  sides ;  still  further,  in  an 
angle  between  two  walls ;  and  to  the  greatest  distance  of  all,  when 
confined  on  four  sides,  and  limited  to  one  direction,  as  in  a  long 
tube.  The  reason  in  these  several  cases  is  obvious ;  for  a  given 
force  can  produce  a  given  amount  of  motion ;  and  if  the  motion  is 
prevented  from  spreading  to  particles  in  some  directions,  it  will 


ACOUSTIC    WAVES.  193 

reach  more  Distant  ones  in  those  directions  in  which  it  does  spread. 
Speaking-tubes  confine  the  movement  to  a  slender  column  of  air, 
and  therefore  convey  sound  to  great  distances,  and  are  on  this  ac- 
count very  useful  in  transmitting  messages  and  orders  between  re- 
mote parts  of  manufacturing  edifices  and  public  houses. 

306.  Nature  of  Acoustic  Waves. — The  vibrations  of  a 
medium  in  the  transmission  of  sound  are  of  the  kind  called  longi- 
tudinal ;  that  is,  the  particles  vibrate  longitudinally  with  regard 
to  the  movement  of  the  sound ;  whereas,  in  water-waves,  the  par- 
ticle-motion is  partly  transverse  to  the  wave-motion  (Art.  247). 
If,  for  example,  sound  is  passing  from  A  to  B  (Fig.  190),  the  par- 

FIG.  190. 
c      r'    c'    r  e  r  c 


tides  just  about  A  are  (at  the  moment  represented)  in  a  state  of 
condensation ;  around  this  condensed  centre  is  a  rarefied  portion, 
then  a  condensed  portion,  &c.,  as  marked  by  the  letters  r,  c,  r',  c, 
r",  &c.  From  r  to  c  the  particles  are  advancing;  so  likewise  from 
r'  to  c',  and  from  r"  to  c".  But  from  c  to  r',  from  c'  to  r',  &c., 
they  are  rebounding.  The  condensed  wave  near  B  has  advanced 
from  A,  and  others  have  followed  it  at  equal  intervals ;  and  be- 
tween these  waves  of  condensation  are  waves  of  rarefaction,  which 
in  like  manner  spread  outward  from  the  centre  A.  And  yet  no 
one  particle  has  any  other  motion  than  a  small  vibration  back  and 
forth  in  the  line,  near  its  original  place  of  rest.  The  amplitude  is 
the  distance  through  which  a  particle  vibrates.  The  intensity  or 
loudness  of  sound  depends  on  the  amplitude. 

In  water-waves  we  distinguish  carefully  between  the  motion 
of  the  ivave  and  the  motion  of  the  water  which  forms  the  wave ; 
so  here,  the  wave-motion  is  totally  different  from  the  motion  of 
the  air  itself.  The  wave,  i.  e.  the  state  of  condensation  and  subse- 
quent rarefaction,  travels  swiftly  forward ;  but  the  masses  of  air, 
which  suffer  these  condensations  and  rarefactions,  simply  tremble 
in  the  line  of  that  motion. 

Since  the  motion  is  propagated  in  all  directions  alike,  the  -en- 
tire system  of  waves  around  the  point  where  sound  originates  con- 
sists of  spherical  strata  of  air  alternately  condensed  and  rarefied. 
As  the  quantity  set  in  motion  in  these  successive  layers  increases 
13 


194  SOUND. 

with  the  square  of  the  distance,  the  amount  of  motion  communi- 
cated to  each  particle  must  diminish  in  the  same  ratio.  Hence, 
the  intensity  of  sound  varies  inversely  as  the  square  of  the  dis- 
tance. 

Fig.  190  is  a  section  of  a  system  of  spherical  waves  around  the 
source  A. 

A  ray  of  sound  is  any  one  of  the  radii  of  the  sphere  whose 
centre  is  the  source  of  sound.  The  vibratory  motion  is  propagated 
along  each  of  the  rays. 

307.  Other  Gaseous  Bodies,  as  Media  of  Sound.— Let 

a  spherical  receiver,  having  a  bell  suspended  in  it,  be  exhausted  of 
air,  till  the  bell  ceases  to  be  heard ;  then  fill  it  with  any  gas  or  va- 
por instead  of  air,  and  the  bell  will  be  heard  again.  By  means  of 
an  organ-pipe  blown  by  different  gases,  it  can  be  learned  with 
what  velocity  sound  would  move  in  each  kind  of  gas  experimented 
upon,  because  the  pitch  of  a  given  pipe  depends  upon  the  velocity 
of  the  waves,  as  will  be  seen  hereafter.  In  hydrogen  sound  is  ex- 
ceedingly feeble,  but  moves  nearly  three  times  as  fast  as  in  air. 
Momentary  development  of  heat  by  compression  produces,  in  all 
gaseous  bodies,  the  effect  of  increasing  the  velocity  of  sound. 

308.  Liquids  as  Media. — Many  experimenters  have  deter- 
mined the  circumstances  of  the  propagation  of  sound  in  water. 
Franklin  found  that  a  person  with  his  head  under  water  could 
hear  the  sound  of  two  stones  struck  together  at  a  distance  of  more 
than  half  a  mile.    In  1826,  Colladon  made  many  careful  experi- 
ments in  the  water  of  Lake  Geneva.    The  results  of  these  and 
other  trials  are  principally  the  following : 

1.  Sounds  produced  in  the  air  are  very  faintly  heard  by  a  per- 
son in  water,  though  quite  near ;  and  sounds  originating  under 
water  are  feebly  communicated  to  the  air  above,  and  in  positions 
somewhat  oblique  are  not  heard  at  all. 

2.  Sounds  are  conveyed  by  water  with  a  velocity  of  4700  feet 
per  second,  at  the  temperature  of  47°  F.,  which  is  more  than  four 
times  as  great  as  in  air.    The  calculated  and  the  observed  velocity 
of  sound  in  water  agree  so  nearly  with  each  other,  that  there  ap- 
pears to  be  no  appreciable  effect  arising  from  heat  developed  by 
compression. 

3.  Sounds  conveyed  in  water  to  a  distance,  lose  their  sonorous 
quality.    For  example,  the  ringing  of  a  bell  gives  a  succession  of 
short  sharp  strokes,  like  the  striking  together  of  two  knife-blajdes. 
The  musical  quality  of  the  sound  is  noticeable  only  within  600  or 
700  feet.    In  air,  it  is  well  known  that  the  contrary  takes  place ; 
the  blow  of  the  bell-tongue  is  heard  near  by,  but  the  continued 
musical  note  is  all  that  affects  the  ear  at  a  distance. 


SOLIDS    AS    MEDIA    OF    SOUND.  195 

4.  Acoustic  shadoivs  are  formed;  that  is,  sound  passes  the 
edges  of  solid  bodies  nearly  in  straight  lines,  and  does  not  turn 
around  them  except  in  a  very  slight  degree.  In  this  respect, 
sound  in  water  resembles  light  much  more  than  it  does  sound  in 
air. 

To  enable  the  experimenter  to  hear  distant  sounds  without 
placing  himself  under  water,  Colladon  pressed  down  a  cylindrical 
tin  tube,  closed  at  the  bottom,  thus  allowing  the  acoustic  pulses  in 
the  water  to  strike  perpendicularly  on  the  sides  of  the  tube.  In 
this  way,  the  faintest  sounds  were  brought  out  into  the  air.  It 
appears  to  be  true  of  sound  as  of  light,  that  it  cannot  pass  from  a 
denser  to  a  rarer  medium  at  large  angles  of  incidence,  but  suffers 
nearly  a  total  reflection. 

309.  Solids  as  Media. — Solid  bodies  of  high  elastic  energy 
are  the  most  perfect  media  of  sound  which  are  known.    An  iron 
rod — as,  for  instance,  a  lightning-rod — will  convey  a  feeble  sound 
from  one  extremity  to  the  other,  with  much  more  distinctness 
than  the  air.    If  the  ears  are  stopped,  and  one  end  of  a  long  wire 
is  held  between  the  teeth,  a  slight  scratch  or  blow  on  the  remote 
end  will  sound  very  loud.    The  sound  in  this  case  travels  through 
the  wire  and  the  bones  of  the  head  to  the  organ  of  hearing.    The 
stethoscope,  an  instrument  used  by  physicians  for  determining 
whether  the  lungs  or  heart  have  a  diseased  or  healthy  action,  illus- 
trates the  conduction  of  sound  by  solids.    The  instrument  is  a 
tubular  rod  of  wood,  one  end  of  which  is  pressed  upon  the  chest 
of  the  patient,  while  the  ear  is  applied  to  the  other.    The  move- 
ments of  the  vital  organs  are  thus  distinctly  heard,  and  the  char- 
acter of  those  movements  readily  distinguished.  .  The  sound  of 
earthquakes  and  volcanic  eruptions  is  transmitted  to  great  dis- 
tances through  the  solid  earth.    By  laying  the  ear  to  the  ground, 
the  tramp  of  cavalry  may  be  heard  at  a  much  greater  distance 
than  through  the  air. 

310.  Velocity   in   Solids.— Structure. — The  velocity  of 
sound  in  cast  iron  is  about  11000  feet  per  second — ten  times 
greater  than  in  air.    This  was  determined  by  Biot,  in  his  experi- 
ments on  some  aqueduct  pipes  in  Paris,  already  alluded  to.    A 
blow  upon  one  end  was  brought  to  an   observer  at  the  other 
through  two  channels,  and  seemed  to  be  two  blows.     One  sound 
traveled  in  the  air  within  the  tube,  the  other  in  the  iron  itself  of 
which  the  pipe  was  made.    From  the  observed  interval  of  time 
between  the  two  sounds,  and  the  known  velocity  of  sound  in  air, 
the  velocity  in  iron  is  readily  calculated.    The  pitch  of  sound  pro- 
duced by  rods  and  tubes  of  different  materials,  when  vibrating 


196  SOUND. 

longitudinally,  enables  us  to  determine  with  tolerable  accuracy  the 
velocity  of  propagation  in  those  substances  respectively. 

In  one  important  particular  solids  differ  from  fluids,  namely, 
in  the  fixed  relations  of  the  particles  among  themselves.  These 
relations  are  usually  different  in  different  directions ;  hence,  sound 
is  likely  to  be  transmitted  more  perfectly  in  some  directions 
through  a  given  solid  than  in  others.  The  scratch  of  a  pin  at 
one  end  of  a  stick  of  timber  seems  loud  to  a  person  whose  ear  is  at 
the  other  end.  The  sound  is  heard  more  perfectly  in  the  direction 
of  the  grain  than  across  it  In  crystallized  substances  it  is  unques- 
tionably true  that  the  vibrations  of  sound  move  with  different 
speed  and  with  different  intensity  in  the  line  of  the  axis,  and  in  a 
line  perpendicular  to  it. 

311.  Mixed  Media. — In  all  the  foregoing  statements  it  has 
been  supposed  that  the  medium  was  homogeneous;  in  other 
words,  that  the  material,  its  density,  and  its  structure,  continue 
the  same,  or  nearly  the  same,  the  whole  distance  from  the  source 
of  sound  to  the  ear.  If  abrupt  changes  occur,  even  a  few  times, 
the  sound  is  exceedingly  obstructed  in  its  progress.  When  the  re- 
ceiver is  set  over  the  bell  on  the  pump  plate,  the  sound  in  the 
room  is  very  much  weakened,  though  the  glass  may  not  be  one- 
eighth  of  an  inch  in  thickness,  and  is  an  excellent  conductor  of 
sound.  The  vibrations  of  the  internal  air  are  very  imperfectly 
communicated  to  the  glass,  and  those  received  by  the,  glass  pass 
into  the  air  again  with  a  diminished  intensity.  If  a  glass  rod  ex- 
tended the  whole  distance  from  the  bell  to  the  ear,  the  sound 
would  arrive  in  less  time,  and  with  more  loudness,  than  if  air 
occupied  the  whole  extent.  For  a  like  reason,  walls,  buildings,  or 
other  intervening  bodies,  though  good  conductors  of  sound  them- 
selves, obstruct  the  progress  of  sound  in  the  air.  This  explains 
the  fact  mentioned  in  Art.  308,  that  sound  in  air  is  heard  faintly 
in  water,  and  vice  versa.  When  the  texture  of  a  substance  is 
loose,  having  many  alternations  of  material,  it  thereby  becomes 
unfit  for  transmitting  sound.  It  is  for  this  reason  that  the  bell- 
stand,  in  the  experiment  just  referred  to,  is  set  on  a  cushion  made 
of  several  thicknesses  of  loose  flannel,  that  it  may  prevent  the  vi- 
brations from  reaching  the  metallic  parts  of  the  pump.  The  waves 
of  sound,  in  attempting  to  make  their  way  through  such  a  sub- 
stance, continually  meet  with  new  surfaces,  and  are  reflected  in  all 
possible  directions,  by  which  means  they  are  broken  up  into  a 
multitude  of  crossing  and  interfering  waves,  and  are  mutually  de- 
stroyed. A  tumbler,  nearly  filled  with  water,  will  ring  clearly ; 
but  if  filled  with  an  effervescing  liquid,  it  will  lose  all  its  sonorous 
quality,  for  the  same  cause  as  before.  The  alternate  surfaces  of  the 
liquid  and  gas,  in  the  foam,  confuse  the  waves,  and  deaden  the  sound. 


REFLECTION    OF    SOUND. 


197 


FIG.  191. 


CHAPTER   II. 

REFLECTION,   REFRACTION,  AND  INFLECTION  OF  SOUND. 

312.  Reflection  of  Sound. — Sound  is  reflected  from  surfaces 
in  accordance  with  the  common  law  of  reflection  in  the  case  of 
elastic  bodies;  that  is, 

T/ie  angle  of  incidence  equals  the  angle  of  reflection,  and  the  two 
angles  are  on  opposite  sides  of  the  perpendicular  to  the  reflecting 
surface. 

Suppose  sound  to  emanate  from  A  (Fig.  191),  and  meet  the 
plane  surface  B  D.  The  particles  of  air  in  the  ray  A  B  vibrate 
back  and  forth  in  that  line, 
and  those  contiguous  to  B 
will,  after  striking  the  sur- 
face, rebound  on  the  line 
B  G,  as  an  elastic  ball  would 
do  (Art.  103),  and  propa- 
gate their  motion  along 
that  line.  The  angle  of 
incidence  A  B  F  equals  the 
angle  of  reflection  F  B  G, 
and  the  two  angles  are  on 
opposite  sides  Q$FB,  which 
is  perpendicular  to  the  re- 
flecting surface  B  D.  If 
G  B  be  produced  back- 
ward, it  will  meet  the  per- 
pendicular A  E  at  Of  as 

far  behind  B  D  as  A  is  before  it.  In  like  manner,  every  ray  of 
sound  after  reflection  proceeds  as  if  from  C,  and  the  successive 
waves  are  situated  as  represented  by  the  dotted  lines  in  the  figure. 
From  the  point  E  the  reflection  is  directly  back  in  the  line  E  A. 

313.  Echoes.— When  sound  is  so  distinctly  reflected  from  a 
surface  that  it  seems  to  come  from  another  source,  it  is  called  an 
echo.    Broad  and  even  surfaces,  such  as  the  walls  of  buildings  and 
ledges  of  rock,  often  produce  this  effect.    According  to  the  law 
(Art.  312),  a  person  can  hear  the  echo  of  his  own  voice  only  by 
standing  in  a  line  which  is  perpendicular  to  the  echoing  surface. 
In  order  that  one  person  may  hear  the  echo  of  another's  voice, 
they  must  place  themselves  in  lines  making  equal  angles  with  the 
perpendicular. 


198  SOUND. 

The  interval  of  time  between  a  sound  and  its  echo  enables  one 
to  judge  of  the  distance  of  the  surface,  since  the  sound  must,  pass 
over  it  twice.  Thus,  if  at  the  temperature  of  74°  the  echo  of  the 
speaker's  voice  reaches  him  in  two  seconds  after  its  utterance,  the 
distance  of  the  reflecting  body  is  about  1130  feet,  and  in  that  pro- 
portion for  other  intervals.  And  he  can  hear  a  distinct  echo  of 
as  many  syllables  as  he  can  pronounce  while  sound  travels  twice 
the  distance  between  himself  and  the  echoing  surface. 

314.  Simple  and  Complex  Echoes. — When  a  sound  is 
returned  by  one  surface,  the  echo  is  called  simple ;  it  is  called 
complex  when  the  reflection  is  from  two  or  more  surfaces  at  differ- 
ent distances,  each  surface  giving  one  echo.    Thus,  a  cannon  fired 
in  a  mountainous  region  is  heard  for  a  long  time  echoed  on  all 
sides,  and  from  various  distances. 

A  complex  echo  may  also  be  produced  by  two  parallel  walls,  if 
the  hearer  and  the  source  of  sound  are  both  situated  between 
them.  The  firing  of  a  pistol  between  parallel  walls  a  few  hundred 
feet  apart  has  been  known  to  return  from  30  to  40  echoes  before 
they  became  too  faint  to  be  heard.  The  rolling  of  thunder  is  in 
part  the  effect  of  reverberation  between  the  earth  and  the  clouds. 
This  is  made  certain  by  the  observed  fact  that  the  report  of  a  can- 
non, which  in  a  level  country  and  under  a  clear  sky  is  sharp  and 
single,  becomes  in  a  cloudy  day  a  prolonged  roar,  mingled  with 
distant  and  repeated  echoes.  But  the  peculiar  inequalities  in  the 
reverberations  of  thunder  are  doubtless  due  in  part  to  the  irregu- 
larly crinkled  path  of  the  electric  spark.  A  discharge  of  lightning 
occupies  so  short  a  time,  that  the  sound  may  be  considered  as 
starting  from  all  points  of  its  track  at  once.  But  that  track  is 
full  of  large  and  small  curves,  some  convex  and  some  concave  to 
the  ear,  and  at  a  great  variety  of  distance ;  and  all  points  which 
are  at  equal  distances  would  be  heard  at  once.  Hence,  the  origi- 
nal sound  comes  to  the  hearer  with  great  irregularity,  loud  at  one 
instant  and  faint  at  another.  These  inequalities  are  prolonged 
and  intensified  by  the  echoes  which  take  place  between  the  clouds 
and  the  earth. 

315.  Concentrated  Echoes. — The  divergence  of  sound  from 
a  plane  surface  continues  the  same  as  before,  that  is,  in  spherical 
waves,  whose  centre  is  at  the  same  distance  behind  the  plane  as 
the  real  source  is  in  front.    But  concave  surfaces  in  general  pro- 
duce a  concentrating  effect.     A  sound  originating  in  the  centre 
of  a  hollow  sphere  will  be  reflected  back  to  the  centre  from  every 
point  of  the  surface.    If  it  emanates  from  one  focus  of  an  ellipsoid, 
it  will,  after  reflection,  all  be  collected  at  the  other  focus.     So,  if 
two  concave  paraboloids  stand  facing  each  other,  with  their  axes 


RESONANCE    OF    ROOMS.  199 

coincident,  and  a  whisper  is  made  at  the  focus  of  one,  it  will  be 
plainly  heard  at  the  focus  of  the  other,  though  inaudible  at  all 
points  between.  In  the  last  case  the  sound  is  twice  reflected,  and 
passes  from  one  reflector  to  the  other  in  parallel  lines.  All  these 
effects  are  readily  proved  from  the  principle  that  the  angles  of  in- 
cidence and  reflection  are  equal. 

The  speaking-trumpet  and  the  ear-trumpet  have  been  supposed 
by  many  writers  to  owe  their  concentrating  power  to  multiplied 
reflections  from  the  inner  surface.  But  a  part  of  the  effect,  and 
sometimes  the  whole,  is  doubtless  due  to  the  accumulation  of  force 
in  one  direction,  by  preventing  lateral  diffusion,  till  the  intensity 
is  greatly  increased.  • 

Concave  surfaces  cause  all  the  curious  effects  of  what  are  called 
whispering  galleries,  such  as  the  dome  of  St.  Paul's,  in  London. 
In  many  of  these  instances,  however,  there  seems  to  be  a  contin- 
ued series  of  reflections  from  point  to  point  along  the  smooth  con- 
cave wall,  which  all  meet  simultaneously  (if  the  curves  are  of  equal 
length)  at  the  opposite  point  of  the  dome;  for  the  whisperer 
places  his  mouth,  and  the  hearer  his  ear,  close  to  the  wall,  and  not 
in  a  focus  of  the  curve.  The  Ear  of  Dionysius  was  probably  a 
curved  wall  of  this  kind  in  the  dungeons  of  Syracuse.  It  is  said 
that  the  words,  and  even  the  whispers,  of  the  prisoners  were  gath- 
ered and  conveyed  along  a  hidden  tube  to  the  apartment  of  the 
tyrant.  The  sail  of  a  ship  when  spread,  and  made  concave  by  the 
breeze,  has  been  known  to  concentrate  and  render  audible  to  the 
sailors  the  sound  of  a  bell  100  miles  distant.  A  concave  shell  held 
to  the  ear  concentrates  such  sounds  as  may  be  floating  in  the  air, 
and  is  suggestive  of  the  murmur  of  the  ocean. 

316.  Resonance  of  Rooms. — If  a  rectangular  room  has 
smooth,  hard  walls,  and  is  unfurnished,  its  reverberations  will  be 
loud  and  long-continued.  Stamp  on  the  floor,  or  make  any  other 
sudden  noise,  and  its  echoes  passing  back  and  forth  will  form  a 
prolonged  musical  note,  whose  pitch  will  be  lower  as  the  apart- 
ment is  larger.  This  is  called  the  resonance  of  the  room.  Now, 
let  furniture  be  placed  around  the  walls,  and  the  reverberations 
will  be  weakened  and  less  prolonged.  Especially  will  this  be  the 
case  if  the  articles  be  of  the  softer  kinds,  and  have  irregular  sur- 
faces. Carpets,  curtains,  stuffed  seats,  tapestry,  and  articles  of 
dress  have  great  influence  in  destroying  the  resonance  of  a  room. 
The  appearance  of  an  apartment  is  not  more  changed  than  is  its 
resonance  by  furnishing  it  with  carpet  and  curtains.  The  blind, 
on  entering  a  strange  room,  can,  by  the  sound  of  the  first  step, 
judge  with  tolerable  accuracy  of  its  size  and  the  general  character 
of  its  furniture. 


200  SOUND. 

The  reason  why  substances  of  loose  texture  do  not  reflect  sound 
well,  is  essentially  the  same  as  what  has  been  stated  (Art  311)  for 
their  not  transmitting  well ;  they  are  not  homogeneous — the  waves 
are  reflected  in  all  directions  by  successive  surfaces,  interfere  with 
each  other,  and  are  destroyed. 

317.  Halls  for  Public  Speaking.— In  large  rooms,  such  as 
churches  and  lecturing  halls,  all  echoes  which  can  accompany  the 
voice  of  the  speaker  syllable  by  syllable,  are  useful  for  increasing 
the  volume  of  sound ;  but  all  which  reach  the  hearers  sensibly 
later,  only  produce  confusion.  It  is  found  by  experiment  that  if 
a  sound  and  its  echo  reach  the  ear  within  one-sixteenth  of  a  second 
of  each  other,  they  seem  to  be  one.  Hence,  this  fraction  of  time 
is  called  the  limit  of  perceptibility.  Within  that  time  an  echo  can 
travel  about  70  feet  more  than  the  original  sound,  and  yet  appear 
to  coincide  with  it.  If  an  echoing  wall,  therefore,  is  within  35 
feet  of  the  speaker,  each  syllable  and  its  echo  will  reach  every 
hearer  within  the  limit  of  perceptibility.  The  distance  may,  how- 
ever, be  increased  to  40  or  even  50  feet  without  injury,  especially 
if  the  utterance  is  not  rapid.  Walls  intended  to  aid  by  their 
echoes  should  be  smooth,  but  not  too  solid;  plaster  on  lath  is 
better  than  plaster  on  brick  or  stone ;  the  first  echo  is  louder,  and 
the  reverberations  less.  Drapery  behind  the  speaker  deprives  him 
of  the  aid  of  just  so  much  echoing  surface.  A  lecturing  hall  is 
improved  by  causing  the  wall  behind  the  speaker  to  change  its  di- 
rection, on  the  right  and  left  of  the  platform,  at  a  very  obtuse 
angle,  so  as  to  exclude  the  rectangular  corners  from  the  room. 
The  voice  is  in  this  way  more  reinforced  by  reflection,  and  there 
is  less  resonance  arising  from  the  parallelism  of  opposite  walls. 
Paneling,  and  any  other  recesses  for  ornamental  purposes,  may 
exist  in  the  reflecting  walls  without  injury,  provided  they  are  not 
curved.  The  ceiling  should  not  be  so  high  that  the  reflection 
from  it  would  be  delayed  beyond  the  limit  of  perceptibility.  Con- 
cave surfaces,  such  as  domes,  vaults,  and  "broad  niches,  should  be 
carefully  avoided,  as  their  effect  generally  is  to  concentrate  all  the 
sounds  they  reflect.  An  equal  diffusion  of  sound  throughout  the 
apartment,  not  concentration  of  it  to  particular  points,  is  the  ob- 
ject to  be  sought  in  the  arrangement  of  its  parts. 

As  to  distant  parts  of  a  hall  for  public  speaking,  the  more  com- 
pletely all  echoes  from  them  can  be  destroyed,  the  more  favorable 
is  it  for  distinct  hearing.  It  is  indeed  true  that  if  a  hearer  is  with- 
in 35  feet  of  a  wall,  however  remote  from  the  speaker,  he  will  hear 
a  syllable,  and  its  echo  from  that  wall,  as  one  sound ;  but  to  all 
the  audience  at  greater  distances  from  the  same  wall,  the  echoes 
will  be  perceptibly  retarded,  and  fall  upon  subsequent  syllables, 


REFRACTION    AND    INFLECTION    OF    SOUND.    201 


thus  destroying  distinctness.  The  distant  walls  should,  by  some 
means,  be  broken*  up  into  small  portions,  presenting  surfaces  in 
different  directions.  A  gallery  may  aid  in  effecting  this ;  and  the 
seats  of  the  gallery  and  of  the  lower  floor  may  rise  rapidly  one  be- 
hind another,  so  that  the  audience  will  receive  directly  much  of  the 
sound  which  would  otherwise  go  to  the  remote  wall,  and  be  re- 
flected. Especially  should  no  large  and  distant  surfaces  be  paral- 
lel to  nearer  ones,  since  it  is  between  parallel  walls  that  prolonged 
reverberation  occurs. 

318.  Refraction  of  Sound. — It  has  been  ascertained  by  ex- 
periment that  sound,  like  light,  may  be  refracted,  or  bent  out  of 
its  rectilinear  course  by  entering  a  substance  of  different  density. 
If  a  large  convex  lens  be  formed  of  carbonic  acid  gas,  by  inclosing 
it  in  a  sphere  of  thin  india-rubber,  a  feeble  sound,  like  the  ticking 
of  a  watch,  produced  on  one  side,  will  be  concentrated  to  a  focal 
point  on  the  other.     In  this  case,  the  several  diverging  rays  of 
sound  are  refracted  toward  each  other  on  entering  the  sphere,  and 
still  more  on  leaving  it,  so  that  they  are  converged  to  a  focus. 

319.  Inflection  of  Sound. — If  air-waves  are  allowed  to  pass 
through  an  opening  in  an  obstructing  wall,  they  are  not  entirely 
confined  within  the  radii  of  the  wave-system  produced  through 
the  opening,  but  spread  with  diminished  intensity  in  lateral  direc- 
tions.    The  particles  near 

the  edges  of  the  opening, 
as  B  and  C  (Fig.  192)  may 
be  considered  as  sources 
of  sound ;  and  if  they  be 
made  centres  of  concentric 
spheres,  whose  radii  are 
equal  to  the  length  of  the 
wave,  B  b,  or  C  c,  and  its 
multiples,  then  these  spher- 
ical surfaces  will  represent 
the  lateral  systems  of 

waves  which  are  diffused  on  every  side  of  the  direct  beam,  B  D, 
C  E.  But  the  sound  is  in  general  more  feeble  as  the  distance 
from  B  D,  or  C  E,  is  greater,  and  in  certain  points  is  destroyed  by 
interference.  This  spreading  of  sound  in  lateral  directions  is 
called  the  inflection  of  sound. 

What  is  true  of  all  sides  of  an  opening  is  of  course  true  when- 
ever sound  passes  by  the  side  of  an  obstacle.  Instead  of  being 
limited  by  lines  almost  straight  drawn  from  the  source,  as  light  is 
in  the  formation  of  a  shadow,  it  bends  round  the  edge,  and  is 


FIG.  192. 


202  SOUND. 

heard,  though  more  feebly,  behind  the  intervening  body.  It  has 
been  already  noticed  (Art.  308)  that  in  water  there  is  little  or  no 
inflection  of  sound. 


CHAPTER    III. 

MUSICAL  SOUNDS  AND  MODES  OF  PRODUCING  THEM. 

320.  The  Vibrations  in  Musical  Sounds.— When  the  im- 
pulses of  a  sounding  body  upon  the  air  are  equidistant,  and  of  suf- 
ficient frequency,  they  produce  what  is  termed  a  musical  sound. 
In  most  cases  these  impulses  are  the  isochronous  vibrations  of  the 
body  itself,  but  not  necessarily  so ;  it  is  found  by  experiment  that 
blows  or  pulses,  of  any  species  whatever,  if  they  are  more  than 
about  15  or  20  per  second,  and  possess  the  property  of  isochronism, 
cause  a  musical  tone.    For  example,  the  snapping  of  a  stick  on 
the  teeth  of  a  metallic  wheel  would  seem  as  unlikely  as  anything 
to  produce  a  musical  sound;    but  when  the  wheel  is  in  rapid 
motion,  the  succession  causes  a  pure  musical  note.    Equidistant 
echoes  often  produce  a  musical  sound,  as  when  a  person  stamps  on 
the  floor  of  a  rectangular  room,  finished,  but  unfurnished  (Art. 
316).    So,  on  a  walk  by  the  side  of  a  long  baluster  fence,  a  sudden 
sharp  sound,  like  the  blow  of  a  hammer  on  a  stone,  brings  back  a 
tone  more  or  less  prolonged,  resembling  the  chirp  of  a  bird.    It  is 
occasioned  by  successive  equidistant  echoes  from  the  balusters  of 
the  fence.    A  flight  of  steps  will  sometimes  produce  the  same 
effect,  the  tone  being  on  a  lower  key  than  that  from  the  fence,  as 
it  should  be. 

321.  The  Pitch  of  Musical  Sounds.— What  is  called  the 
pitch  of  a  musical  sound,  or  its  degree  of  acuteness,  is  owing  en- 
tirely to  its  rate  of  vibration.    Other  qualities  of  sounds  are  due 
to  other  and  often  unknown  circumstances ;  but  rapidity  of  vibra- 
tion is  the  only  condition  on  which  the  pitch  depends.    In  compar- 
ing one  musical  sound  with  another,  if  the  number  of  vibrations 
per  second  is  greater,  the  sound  is  more  acute,  and  is  said  to  be  of 
a  higher  pitch ;  if  the  vibrations  are  fewer  per  second,  the  sound 
is  graver,  or  of  a  lower  pitch. 

322.  The  Monochord. — If  a  string  of  uniform  size  and  text- 
ure is  stretched  on  a  box  of  thin  wood,  by  means  of  a  pulley  and 
weight,  the  instrument  is  called  a  monochord,  and  is  useful  for 
studying  the  laws  of  vibrations  in  musical  sounds.    The  sound 


THE    MONOCHORD. 

emitted  by  the  vibrations  of  the  whole  length  of  the  string  is 
called  its  fundamental  sound. 

If  the  string  be  drawn  aside  from  its  straight  position,  and 
then  released,  one  component  of  the  force  of  tension  urges  every 
particle  back  towards  its  place  of  rest ;  but  the  string  passes  be- 
yond that  place,  on  account  of  the  momentum  acquired,  and  de- 
viates as  far  on  the  other  side;  from  which  position  it  returns,  for 
the  same  reason  as  before,  and  continues  thus  to  vibrate  till  ob- 
structions destroy  its  motion.  By  the  use  of  a  bow,  the  vibrations 
may  be  continued  as  long  as  the  experimenter  chooses. 

The  pitch  of  the  fundamental  sound  of  musical  strings  is  found 
by  experience  to  depend  on  three  circumstances;  the  length  of  the 
string — its  weight  or  quantity  of  matter — and  its  tension.  The 
tone  becomes  more  acute  as  we  increase  the  tension,  or  diminish 
either  the  length  or  the  weight.  The  operation  of  these  several 
circumstances  may  be  seen  in  a  common  violin.  The  pitch  of  any 
one  of  the  strings  is  raised  or  lowered  by  turning  the  screw  so  as 
to  increase  or  lessen  its  tension ;  or,  the  tension  remaining  the 
same,  higher  or  lower  notes  are  produced  by  the  same  string,  by 
applying  the  fingers  in  such  a  manner  as  to  shorten  or  lengthen 
the  string  which  is  vibrating ;  or,  both  the  tension  and  the  length 
of  the  string  remaining  the  same,  the  pitch  is  altered  by  making 
the  string  larger  or  smaller,  and  thus  increasing  or  diminishing  its 
weight. 

A  string  is  said  to  make  a  single  vibration  in  passing  from  the 
extreme  limit  on  one  side  to  the  extreme  limit  on  the  other ;  a 
double  vibration  is  the  motion  across  and  back  again  to  the  origi- 
nal position.  Independently  of  calculation,  it  is  easy  to  see  that, 
with  a  given  weight  per  inch,  and  a  given  tension,  the  string  will 
vibrate  slower,  if  longer,  since  there  is  more  matter  to  be  moved, 
and  only  the  same  force  to  move  it ;  and  for  a  similar  reason,  the 
length  and  tension  being  given,  it  will  also  vibrate  slower,  if 
heavier.  On  the  other  hand,  if  length  and  weight  are  given,  it 
will  vibrate  faster,  if  the  tension  is  greater  ;  because  a  greater  force 
will  move  a  given  quantity  at  a  swifter  rate. 

323.  Time  of  a  Single  Vibration. — The  mathematical 
formula  for  the  time  of  a  vibration  is  the  following,  in  which  T  = 
the  time  of  a  single  vibration ;  I  =  the  length  of  the  string  in 
inches ;  w  —  the  weight  of  one  inch  of  the  string ;  i  =  the  ten- 
sion in  Ibs. ;  and  g  =  the  force  of  gravity  =  386  inches  =  32 1 
feet  (Art.  28) ; 


204  SOUND. 

The  constant  factor,  g,  being  omitted,  the  variation  may  be  ex- 
pressed thus : 

T  oc  — =r- ;  that  is, 
Vt 

TJie  time  of  a  vibration  varies  as  the  length  of  the  string  multi- 
plied by  the  square  root  of  its  weight  per  inch,  and  divided  by  the 
square  root  of  its  tension. 

As  the  distance  of  the  string  from  its  quiescent  position  does 
not  form  an  element  of  the  algebraic  expression  for  the  time  of  a 
vibration,  it  follows  that  the  time  is  independent  of  the  amplitude. 
Hence,  as  in  the  pendulum,  the  vibrations  of  a  string,  fixed  at 
both  ends,  are  performed  in  equal  times,  whether  the  amplitude 
of  the  vibrations  be  greater  or  smaller.  It  is  on  this  account  that 
the  pitch  of  a  string  does  not  alter,  when  left  to  vibrate  till  it 
stops.  The  excursions  from  side  to  side  grow  less,  and  therefore 
the  sound  more  feeble,  till  it  ceases ;  but  the  rate  of  vibration,  and 
therefore  the  pitch,  remains  the  same  to  the  last.  This  property 
of  isochronism,  independent  of  extent  of  excursion,  is  common  to 
sounding  bodies  generally,  and  is  owing  to  what  may  be  called  the 
law  of  elasticity,  that  the  restoring  force,  acting  on  any  particle, 
varies  directly  as  its  distance  from  the  place  of  rest.  For  example, 
each  particle  of  the  string,  if  removed  twice  as  far  from  its  place 
of  rest,  is  urged  back  by  a  force  twice  as  great,  and  therefore  re- 
turns in  the  same  time. 

324.  The  Number  of  Vibrations  in  a  Given  Time. — The 

greater  is  the  length  of  one  vibration,  the  less  will  be  the  number 
of  vibrations  in  a  given  time ;  that  is,  if  N  represents  the  number, 

,_       1     .  -,      I  Vw       ,r       Vt       T«  .      , 

N  x  7=-;  but  as  T  x  — — ,  .*.  N  x  — — .    If  t  and  w  are  con- 

T'  Vt  iVw 

stant,  N  x  T ;  if  I  and  t  are  constant,  N  x  — — ;  and  if  I  and  w 
<>  V  w 

are  constant,  N  x  Vt ;  that  is, 

1.  The  number  of  vibrations  varies  inversely  as  the  length. 

2.  The  number  of  vibrations  varies  inversely  as  the  square  root 
of  the  weight  of  the  string. 

3.  The  number  of  vibrations  varies  as  the  square  root  of  the 
tension. 

Thus,  the  number  of  vibrations  in  a  second  may  be  doubled, 
either  by  halving  the  length  of  the  string  or  by  making  its  weight 
one-fourth  as  great,  or,  finally,  by  making  its  tension  four  times  as 
great. 

325.  Vibrations  of  a  String  in  Parts. — The  monochord 
may  be  made  to  vibrate  in  parts,  the  points  of  division  remaining 


VIBRATIONS    OF    A    STRING    IN    PARTS.          205 

at  rest ;  and  this  mode  of  vibration  may  even  coexist  with  the  one 
already  described.  Of  course  the  sound  produced  by  the  parts 
will  be  on  a  higher  pitch,  since  they  are  shorter,  while  the  tension 
and  the  weight  per  inch  remain  unaltered.  It  is  a  noticeable  fact 
that  the  parts  are  always  such  as  will  exactly  measure  the  whole 
without  a  remainder.  Hence  the  vibrating  parts  are  either  halves, 
thirds,  fourths,  or  other  aliquot  portions.  The  sounds  produced 
by  any  of  these  modes  of  vibration  are  called  harmonics,  for  a  rea- 
son which  will  appear  hereafter.  Suppose  a  string  (Fig.  193)  to 

FIG.  193. 


be  stretched  between  A  and  B,  and  that  it  is  thrown  into  vibra- 
tion in  three  parts.  Then  while  A  D  makes  its  excursion  on  one 
side,  D  C  will  move  in  the  opposite  direction,  and  C  B  the  same 
as  A  D\  and  when  one  is  reversed,  the  others  are  also,  as  shown 
by  the  dotted  line.  In  this  way  D  and  C  are  kept  at  rest,  being 
urged  toward  one  side  by  one  portion  of  string,  and  toward  the 
opposite  by  the  next  portion.  But  the  string  may  at  the  same 
time  vibrate  as  a  whole  ;  in  which  case  D  and  C  will  have  motion 
to  each  side  of  their  former  places  of  rest,  while  relatively  to  them 
the  three  portions  will  continue  their  movements  as  before.  The 
points  C  and  D  are  called  nodes ;  the  parts  A  D,  D  C,  and  C  B, 
are  called  ventral  segments.  By  a  little  change  in  the  quickness 
of  the  stroke,  the  bow  may  be  made  to  bring  from  the  monochord 
a  great  number  of  harmonic  notes,  each  being  due  to  the  vibra- 
tions of  certain  aliquot  parts  of  the  string.  By  confining  a  partic- 
ular point,  however,  at  the  distance  of  ±,  |,  or  other  simple  fraction 
of  the  whole  from  the  end,  the  particular  harmonic  belonging-  to 
that  mode  of  division  may  be  sounded  clear,  and  unmmgled  with 
the  others. 

326.  Vibrations  cf  a  Column  of  Air. — When  a  musical 
sound  is  produced  by  a  pipe  of  any  kind,  it  is  the  column  of  in- 
closed air  which  must  be  regarded  as  the  sounding  body.  A  con- 
densed wave  is  caused,  by  some  mode  of  excitation,  to  travel  back 
and  forth  in  the  pipe,  followed  by  a  rarefied  portion ;  and  these 
waves  affect  the  surrounding  air  much  in  the  same  way  as  do  the 
alternate  excursions  of  a  string.  That  it  is  the  air,  and  not  the 
pipe  itself,  which  is  the  source  of  sound,  is  proved  by  using  pipes 
of  various  materials — the  most  elastic  and  the  most  inelastic — as 
glass,  wood,  paper,  and  lead ;  if  they  are  of  the  same  form  and  size, 
the  tone  in  each  case  has  the  same  pitch. 

In  order  to  examine  the  manner  in  which  the  air-columns  in 


206  SOUND. 

pipes  perform  their  vibrations,  it  is  convenient  to  consider  them 
in  three  classes : 

1st.  Pipes  which  are  closed  at  both  ends. 

3d.  Those  which  are  closed  at  one  end  and  open  at  the  other. 

3d.  Those  which  are  open  at  both  ends. 

327.  Both  Ends  of  the  Pipe  Closed.— Suppose  the  ends 
of  the  pipe,  AC  B  (Fig.  194),  to  be  closed,  and  an  impulse  in  some 
way  to  be  communicated 

at  the  centre,  (7;  then  the  Frq- ^ 

motion  of  the  column  will 
consist  of  a  constant  and 
regular  fluctuation  of  the 
whole  mass  to  and  fro  with- 
in the  pipe,  the  air  being  always  condensed  in  one  half,  while  it  is 
rarefied  in  the  other.  While  the  condensed  pulse  moves  from  B 
to  Of  the  point  of  rarefaction  runs  from  A  to  (7,  where  they  pass 
each  other ;  hence,  at  the  middle  of  the  pipe  there  is  no  change  of 
density,  since  every  degree  of  condensation  is  at  that  point  met  by 
an  equal  degree  of  rarefaction  of  the  other  half  of  the  general 
wave.  At  the  extremities,  A  and  B,  there  is  alternately  a  maxi- 
mum of  condensation  and  of  rarefaction,  each  being  reflected  and 
returning,  to  meet  again  at  O.  Fig.  195  shows  the  air  in  a  state 
of  condensation  at  A,  and 

of  rarefaction  at  B.    At  FlG- 195- 

all  points  between  the  cen- 
tre and  the  ends  there  is 
alternate  condensation  and 
rarefaction,  but  in  a  less 
degree  according  to  the  distance  from  the  ends. 

On  the  other  hand,  the  excursions  of  the  particles  are  greatest 
at  (7,  and  nothing  at  A  and  B,  where  all  motion  is  prevented  by 
the  fixed  stoppers  by  which  the  pipe  is  closed.  Between  the  ends 
and  the  centre,  the  amplitude  of  vibration  is  greater,  as  the  dis- 
tance from  the  centre  is  less. 

The  pitch  of  such  a  pipe  will  be  lower,  as  the  pipe  is  longer, 
because  the  waves  have  a  greater  distance  to  travel  between  the 
successive  reflections,  and  hence  there  will  be  a  smaller  number 
per  second.  So  also,  lowering  the  temperature  lowers  the  pitch, 
since  the  wave  then  travels  more  slowly,  and  suffers  fewer  reflec- 
tions in  a  second. 

328.  One  End  of  the  Pipe  Closed,  the  other  Open  —If, 

while  the  column  A  B  is  vibrating  as  a  whole,  an  aperture  is  made 
at  the  centre,  or  even  if  the  pipe  is  divided  there,  so  that  the  aper- 
ture extends  entirely  round  it,  this  will  not  interrupt  the  oscilla- 


VIBRATIONS    IN    PIPES.  207 

tion  already  described,  because  there  is  neither  rarefaction  nor 
condensation  at  the  point  C,  and  hence  no  tendency  there  to  lat- 
eral motion.     The  means  employed  for  ex- 
citing vibrations  may  therefore  be  applied  FIG.  196.          ^ 
at  the  open  section.    Let  the  pipe  A  B  (Fig. 
196),  remaining  stopped  at  A  and  B,  be  di- 
vided at  the  middle,  <7;  and  let  the  half 
pipe  B  C  be  removed,  while  the  exciting 

cause  remains  at  C  (Fig.  194),  then  the  vibrations  in  A  C  will  still 
continue,  and  the  pitch  be  unaltered.  For  now  the  condensed 
pulse,  on  reaching  C,  will  be  returned  to  A  by  the  vibrating  disk 
or  spring  which  excites  it,  and  will  make  a  second  reflection  at  A 
at  the  same  instant  as  it  would  have  done  at  B  in  the  whole  pipe 
A  B\  thus  the  same  movements  are  performed  now  in  one  half 
which  were  before  performed  alternately  in  the  two.  Hence  it  is 
that  a  pipe  with  only  one  end  closed,  and  a  pipe  of  twice  its  length, 
with  both  ends  closed,  give  the  same  pitch. 

329.  Both  Ends  of  the  Pipe  Open.— When  both  ends  of 
a  pipe  are  open,  it  may  still  produce  a  musical  tone,  by  having  a 
node  in  the  centre  of  it,  thus  forming  two  pipes  like  the  one  last 
described.    When  the  vibration  is  established  in  such  a  pipe,  the 
pulses  from  the  ends  move  simultaneously  toward  C  (Fig.  197)) 
and  again  from  it  after  re- 
flection.   Thus  C  is  a  fixed  FlG- 197- 

point,  where  the  greatest 
condensation  and  rarefac- 
tion occur  alternately,  like      ^ 
A  in  Fig.  194.    It  there- 
fore has  the  same  pitch  as  A  C  alone,  stopped  at  C  and  open  at  A. 
If  a  solid  partition  be  inserted  at  (7,  it  causes  no  change  of  pitch. 
Such  a  pipe  can  produce  no  sound,  except  by  the  formation  of 
at  least  one  node. 

330.  The  Second  Kind   of  Pipe   is   the   Elementary 
Form. — In  comparing  with  each  other  the  three  kinds  of  pipe 
which  have  been  described,  it  is  observable  that  the  first  kind 
(stopped  at  both  ends),  and  the  third  kind  (open  at  both  ends),  is 
each  a  double  pipe  of  the  second  kind  (open  at  one  end,  and 
stopped  at  the  other).    For,  if  two  pipes  of  the  second  kind  be 
placed  with  their  open  ends  together,  as  we  have  seen,  they  form 
one  of  the  first  kind,  and  there  is  no  change  of  pitch.    Again,  if 
the  two  be  placed  with  the  closed  ends  in  contact,  they  form  a  pipe 
of  the  third  class ;  since  the  partition  may  remain  or  be  removed, 
without  affecting  the  mode  of  vibration.    Hence,  a  pipe  open  at 
both  ends,  and  one  of  the  same  length  closed  at  both  ends,  each 


208  SOUND. 

yields  the  same  fundamental  note  as  a  pipe  of  half  their  length, 
open  only  at  one  end. 

331.   Vibrations  of  a   Column  of  Air   in   Parts. — The 

same  is  true  of  a  column  of  air  as  of  a  string,  that  it  may  vibrate 
in  parts ;  and  also,  that  two  or  more  modes  of  vibration  may  co- 
exist in  the  same  column. 

The  first  and  third  kinds  of  pipe  can  divide  so  that  the  whole 
and  the  vibrating  segments  have  the  ratios  of  1 :  4  :  4  :  &c. ;  these 
ratios  in  the  closed  pipe  are  shown  in  Figs.  194, 198,  and  199 ;  and 
in  the  open  pipe  in  Figs.  197  and  200.  In  Fig.  198  the  pipe  is 
divided  into  two  equal 

parts,  in  each  of  which  Fm- 198- 

the  vibrations  take  place 
in  the  same  manner  as  in 
the  whole,  Fig.  194. 
Condensations  run  si- 
multaneously from  A  and  B  to  the  middle  point  C,  and  thence 
back  to  A  and  B.  When  C  is  condensed,  A  and  B  are  rarefied ; 
and  when  A  and  B  are  condensed,  C  is  rarefied.  Those  three 
points  have  no  amplitude,  but  the  greatest  changes  in  density. 
But  the  points  midway  between  have  the  greatest  amplitude,  and 
no  change  of  density.  As  the  waves  run  over  the  parts  in  half 
the  time  that  they  would  over  the  whole,  the  pitch  is  raised  accord- 
ingly. In  this  mode  of  vibrating,  the  opening  where  the  vibra- 
tions are  excited  cannot  be  at  C,  where  the  node  is  formed. 

In  Fig.  199  are  shown 

three  vibrating  segments.  FlG- 199- 

B  and  D  are  condensed  at 
one  moment,  A  and  E  at 
another. 

In  the  third  kind,  as 
already  stated  (Art.  329),  there  must  be  at  least  one  node.  When 
there  are  two,  it  is  apparent  by  Fig.  200  that  they  must  be  one- 
fourth  of  the  length  from 

each  end,  in  order  that  the     FiG.200^ 

three  parts  may  vibrate  in 
unison:  for  the  middle 


C 


part  is  a  complete  segment,     :r~          c  E  Jt  JS 

like  the  pipe  A  B  (Fig. 

194),  while  the  ends  are  half  segments,  like  the  pipe  A  C  (Fig. 
196).  If  there  were  three  nodes,  there  would  be  two  complete 
segments  between  them,  and  two  half  segments  at  the  ends.  It  is 
evident  that  the  lengths  of  the  half  segments,  being  J,  -[,  J,  &c., 
are  as  1,  J,  i,  &c.,  of  the  whole  pipe ;  therefore  the  rates  of  vibra- 


MODES    OF    EXCITING    THEM. 


209 


FIG.  201 . 


tion  (being  inversely  as  the  lengths)  are  as  the  numbers  1,  2,  3, 
&c. 

In  the  second  kind  of  pipe  the  ratios  of  length  for  successive 
modes  of  vibration  are  1  :  J  :  I,  &c.  The  simplest  division  is  by 
one  node,  a  third  of  the  length  from  the  open  end,  as  in  Fig.  201. 
Then  0  D,  a  half  segment,  and  A  D,  a  com- 
plete segment,  have  the  same  rate  of  vibra- 
tion. If  there  were  two  nodes,  one  must 
be  a  fifth  from  the  open  end,  while  the 
other  divides  the  remainder  into  two  com-  __  _ 

plete  segments.     Therefore,  in  the  several 

modes  of  vibration  of  the  second  kind  of  pipe,  the  half  segments, 
being  1,  |,  |,  &c.,  of  the  whole  length,  the  rates  of  vibration  in 
them  are  as  the  odd  numbers  1,  3,  5,  &c. 

332.  Modes  of  Exciting  Vibrations  in  Pipes.. — There 
are  two  methods  of  making  the  air-column  in  a  pipe  to  vibrate: 
one  by  a  stream  of  air  blown  across  an  orifice  in  the  pipe,  the 
other  by  an  elastic  plate  called  a  reed.  A  familiar  example  of  the 
first  is  the  flute.  A  stream  of  air  from  the  lips  is  directed  across 
the  embouchure,  so  as  just  to  strike  the  opposite  edge ;  this  causes 
a  wave  to  move  through  the  tube.  The  stream  of  air,  like  a  spring, 
vibrates  so  as  to  keep  time  with  the  movement  of  the  wave  to  and 
fro,  while  at  each  pulse  it  renews  that  movement,  and  makes  the 
sound  continuous.  For  higher  notes,  the  stream  must  be  blown 
more  swiftly,  that  by  its  greater  elastic  force,  it  may  be  able  to 
conform  to  the  more  rapid  vibration  of  the  column.  A  large  pro- 
portion of  the  pipes  of  an  organ  are  made  to  produce  musical 
tones  essentially  in  the  same  way  as  the  flute,  and  are  called  mouth- 


Fig.  202  shows  the  construction  of  the  mouth-pipe  of  an  organ ; 
o  b  is  the  mouth ;  and  as  the  stream  of  air  issues  from      FJ(J  2Q3 
the  channel  i,  it  starts  a  wave  in  the  pipe,  and  then 
the  stream  itself  vibrates  laterally  past  the  lip  #,  keep- 
ing time  with  the  successive  returns  of  the  wave  in 
the  pipe.     The  pipe  is  attached  to  the  wind-chest  by 
the  foot  P. 

The  clarinet  is  an  example  of  vibrations  in  an  air- 
column  by  a  reed.  In  that  instrument  the  reed  is 
often  made  of  wood ;  when  the  air  is  blown  past  its 
edge  into  the  tube,  the  reed  is  thrown  into  vibration, 
and  by  it  the  column  of  air.  The  strength  of  elasticity 
in  the  reed  should  be  such  that  its  vibrations  will  keep 
time  with  the  excursions  of  the  wave  in  the  column. 
What  are  called  the  reed  pipes  of  the  organ  are  con- 
14 


210 


SOUND. 


structed  on  the  same  principle,  but  the  reeds  are  metallic.  An  ex- 
ample is  seen  in  Fig.  203,  which  represents  a  model  of  the  reed 
pipe,  made  to  show  the  vibrations  through  the 
glass  walls  at  E.  A  chimney,  H,  is  usually  FlG- 

attached,  sometimes  of  a  form  (as  in  the  figure) 
to  increase  the  loudness  of  the  sound,  and 
sometimes  of  a  different  form,  for  softening  it. 

333.  Vibrations  of  Rods  and  Lam- 
inae.— A  plate  of  metal  called  a  reed  is  much 
used  for  musical  purposes  in  connection  with 
a  column  of  air,  as  already  stated.     Except  in 
such  connection,  the  sounds  of  wires  and  lam- 
inse  are  generally  too  feeble  to  be  employed  in 
music.    But  their  vibrations  have  been  much 
studied,  on  account  of  the  interesting  phe- 
nomena attending  them. 

334.  Wires. — If  one  end  of  a  steel  wire 
is  fastened  in  a  vise  and  vibrated,  wbile  a  thin 
blade  of  lunlight  falls  across  it,  the  path  of  the 
illuminated  point  may  be  traced.    It  is  not 
ordinarily  a  circular  arc  about  the  fixed  point 
as  a  centre,  but  some  irregular  figure;  and 

frequently  the  point  describes  two  systems  of  ellipses,  the  vibra- 
tions passing  alternately  from  one  system  to  the  other  several 
times  before  running  down.  If  the  structure  of  the  wire  were  the 
same  in  every  part  across  its  section,  and  if  the  fastening  pressed 
equally  on  every  point  around  it,  the  orbit  of  each  particle  would 
be  a  series  of  ellipses,  whose  major  axes  are  on  the  same  line.  If, 
moreover,  there  was  no  obstruction  to  the  motion,  and  the  law  of 
elasticity  could  obtain  perfectly,  it  would  vibrate  in  the  same 
elliptic  orbit  forever,  the  force  toward  the  centre  being  directly 
as  the  distance.  It  is  easy  to  cause  the  wire,  in  the  experiment 
just  described,  to  vibrate  also  in  parts ;  in  which  case  each  atom, 
while  describing  the  elliptic  orbit,  will  perform  several  smaller 
circuits,  which  appear  as  waves  on  the  circumference  of  the  larger 
figure. 

335.  Chladni's  Plates.— If  a  square  plate  of  glass  or  elastic 
metal,  of  uniform  thickness  and  density,  be  fastened  by  its  centre 
in  a  horizontal  position,  and  a  bow  be  drawn  on  its  edge,  it  will 
emit  a  pure  musical  tone ;  and  by  varying  the  action  of  the  bow, 
and  touching  different  points  of  the  edge  with  the  finger,  a  variety 
of  sounds  may  be  obtained  from  it.    The  plate  necessarily  vibrates 
in  parts ;  and  the  lowest  pitch  is  produced  when  there  are  two 


CHLADNI'S    PLATES.      BELLS.  211 

nodal  lines  parallel  to  the  sides,  and  crossing  at  the  centre,  thus 
dividing  the  plate  into  four  square  ventral  segments.  The  posi- 
tion of  the  nodal  lines,  and  the  forms  of  the  segments,  are  beauti- 
fully exhibited  by  sprinkling  writing-sand  on  the  plate.  The  par- 
ticles will  dance  about  rapidly  till  they  find  the  lines  of  rest,  where 
they  will  presently  be  collected.  For  every  new  tone  the  sand  will 
show  a  new  arrangement  of  nodal  lines;  and  as  two  or  more 
modes  of  vibration  may  coexist  in  plates,  as  well  as  in  strings  and 
columns  of  air,  the  resultant  nodes  will  also  be  rendered  visible. 
Again,  by  fastening  the  plate  at  a  different  point,  still  other  ar- 
rangements will  take  place,  each  distinguishable  by  the  position 
of  its  nodal  lines  and  the  pitch  of  its  musical  note.  The  form  of 
the  plate  itself  may  also  be  varied,  and  each  form  will  be  charac- 
terized by  its  own  peculiar  systems.  Chladni,  who  first  performed 
these  interesting  experiments,  delineated  and  published  the  forms 
of  ninety  different  systems  of  vibration  in  the  square  plate  alone. 

If  a  fine  light  powder,  as  lycopodium  (the  pollen  of  a  species 
of  fern),  be  scattered  on  the  plate,  it  is  affected  in  a  very  different 
manner  from  heavy  sand.  It  will  gather  into  rounded  heaps  on 
those  portions  of  the  segments  which  have  the  greatest  amplitude 
of  vibration ;  the  particles  which  compose  the  heaps  performing  a 
continual  circulation,  down  the  sides  of  the  heaps  along  the  plate 
to  the  centre,  and  up  the  axis.  If  the  vibration  is  violent,  the 
heaps  will  be  thrown  up  from  the  plate  in  little  clouds  over  the 
portions  of  greatest  motion.  The  cause  of  this  singular  effect  was 
ascertained  by  Faraday,  who  found  that  in  an  exhausted  receiver 
the  phenomenon  ceased.  It  is  due  to  a  circulation  of  the  air, 
which  lies  in  contact  with  a  vibrating  plate.  The  air  next  to 
those  parts  which  have  the  greatest  amplitude  is  at  each  vibration 
thrown  upward  more  powerfully  than  elsewhere,  and  surrounding 
particles  press  into  its  place,  and  thus  a  circulation  is  established ; 
and  a  fine  light  powder  is  more  controlled  by  these  atmospheric 
movements  than  by  the  direct  action  of  the  plate. 

336.  Bells. — If  a  thin  plate  of  metal  takes  the  form  of  a  cyl- 
inder or  bell,  its  fundamental  note  is  pro-  -. 
duced  when  each  ring  of  the  material 
changes  from  a  circle  to  an  ellipse,  and 
then  into  a  second  ellipse,  whose  axis  is 
at  right  angles  to  the  former,  as  seen  in 
Fig.  204.  It  thus  has  four  ventral  seg- 
ments and  four  nodal  lines,  the  latter  ly- 
ing in  the  plane  of  the  axis  of  the  bell  or 
cylinder.  If  the  rings  which  compose 
the  bell  were  all  detached  from  one  an- 


SOUND. 

other,  they  would  have  different  rates  of  vibration  according  to 
their  diameter,  and  hence  would  produce  tones  of  various  pitch ; 
but,  being  bound  together  by  cohesion,  they  are  compelled  to  keep 
the  same  time,  and  hence  give  but  one  fundamental  tone.  But  a 
bell,  especially  if  quite  thin,  may  be  made  to  emit  a  series  of  har- 
monic sounds  by  dividing  up  into  a  greater  number  of  segments. 
It  is  obvious  that  the  number  of  nodes  must  always  be  even,  be- 
cause two  successive  segments  must  move  in  opposite  directions 
in  one  and  the  same  instant ;  otherwise  the  point  between  them 
could  not  be  kept  at  rest,  and  therefore  would  not  be  a  node.  Be- 
sides the  principal  tone  of  a  church-bell,  one  or  two  subordinate 
sounds  on  a  different  pitch  may  usually  be  detected.  A  glass  bell, 
suitably  mounted  for  the  lecture-room,  will  yield  ten  or  twelve 
harmonics,  by  means  of  a  bow  drawn  on  its  edge. 

337.  The  Voice. — The  vocal  organ  is  complex,  consisting  of 
a  cavity  called  the  larynx,  and  a  pair  of  membranous  folds  like 
valves,  having  a  narrow  opening  between  them ;  this  opening, 
called  the  glottis,  admits  the  air  to  the  larynx  from  the  wind-pipe 
below.  The  edges  of  these  valves  are  thickened  into  a  sort  of 
cord,  and  for  this  reason  the  apparatus  is  called  the  vocal  cords.  In 
the  act  of  breathing,  the  folds  of  the  glottis  lie  relaxed  and  sepa- 
rate from  each  other,  and  the  air  passes  freely  between  them,  with- 
out producing  vibration.  But  in  the  effort  to  form  a  vocal  sound, 
they  approach  each  other,  and  become  tense,  so  that  the  current 
of  air  throws  them  into  vibration.  These  vibrations  are  enforced 
by  the  consequent  vibrations  in  the  air  of  the  larynx  above ;  and 
thus  a  fullness  of  sound  is  produced,  as  in  many  musical  instru- 
ments, in  which  a  reed,  and  the  air  of  a  cavity,  perform  synchro- 
nous vibrations,  and  emit  a  much  louder  sound  than  either  could 
do  alone.  If  two  pieces  of  thin  india-rubber  be  stretched  across 
the  end  of  a  tube,  with  their  edges  parallel,  and  separated  by  a 
narrow  space,  as  represented  in  Fig.  205,  the  ar- 
rangement will  give  an  idea  of  the  larynx  and 
glottis  of  the  vocal  organ.  If  air  be  forced  through, 
a  sound  is  produced,  whose  pitch  depends  on  the 
size  of  the  tube  and  the  tension  of  the  valves. 

The  natural  key  of  a  person's  voice  depends  on 
the  length  and  weight  of  the  vocal  cords,  and  the 
size  of  the  larynx.  The  yielding  nature  of  all  the 
parts,  and  the  ability,  by  muscular  action,  to  change 
the  form  and  size  of  the  cavity  and  the  tension  of  the  valves,  give 
great  variety  to  the  pitch,  and  the  power  of  adjusting  it  with  pre- 
cision to  every  shade  of  sound  within  certain  limits.  No  instru- 
ment of  human  contrivance  can  be  brought  into  comparison  with 


TIIE    ORGAN    OF    HEARING. 


213 


the  organ  of  voice.  After  the  voice  is  formed  .by  its  appropriate 
organ,  it  undergoes  various  modifications,  by  means  of  the  palate, 
the  tongue,  the  teeth,  the  lips,  and  the  nose,  before  it  is  uttered  in 
the  form  of  articulate  speech. 

338.  The  Organ  of  Hearing. — The  principal  parts  of  the 
ear  are  the  following  : 

1.  The  outer  ear,  E  a  (Pig.  206),  terminating  at  the  membrane 
of  the  tympanum,  m.   , 

FIG.  206. 


2.  The  tympanum,  a  cavity  separated  from  the  outer  ear  by  a 
membrane,  m,  and  containing  a  series  of  four  very  small  bones 
(ossicles),  b,  c,  o,  and  s,  severally  called,  on  account  of  their  form, 
the  hammer,  the  anvil,  the  Ml,  and  the  stirrup.     The  figure  rep- 
resents the  walls  of  the  tympanum  as  mostly  removed,  in  order  to 
show  the  internal  parts.    This  cavity  is  connected  with  the  back 
part  of  the  mouth  by  the  Eustachian  tube,  d. 

3.  The  labyrinth,  consisting  of  the  vestibule,  v,  the  semicircular 
canals,  f,  and  the  cochlea,  g.     The  latter  is  a  spiral  tube,  winding 
two  and  a  half  times  round.     The  parts  of  the  labyrinth  are  exca- 
vated in  the  hardest  bone  of  the  body.     The  figure  shows  only  its 
exterior.    There  are  two  orifices  through  the  bone  which  separates 
the  labyrinth  from  the  tympanum,  the  round  orifice,  e,  passing 
into  the  cochlea,  and  the  oval  orifice,  s,  leading  to  the  vestibule. 
These  orifices  are  both  closed  by  a  thin  membrane.    The  ossicles 
of  the  tympanum  form  a  chain  which  connects  the  centre  of  the 
membrane,  m,  with  that  which  closes  the  oval  orifice.    The  laby- 
rinth is  filled  with  a  liquid,  in  various  parts  of  which  float  the 
fibres  of  the  auditory  nerve. 

By  the  form  of  the  outer  ear,  the  waves  are  concentrated  upon 
the  membrane  of  the  tympanum,  thence  conveyed  through  the 


214:  SOUND. 

chain  of  bones  to  the  membrane  of  the  labyrinth,  and  by  that  to 
the  liquid  within  it,  and  thus  to  the  auditory  nerve,  whose  fibres 
lie  in  the  liquid. 


CHAPTER   IY. 

MUSICAL  SCALES.— THE  RELATIONS  OF  MUSICAL   SOUNDS. 

339.  Numerical  Relations  of  the  Notes.— To  obtain  the 
series  of  notes  which  compose  the  common  scale  of  music,  it  is 
convenient  to  use  the  monochord.     Calling  the  sound,  which  is 
given  by  the  whole  length  of  the  string,  the  fundamental,  or  key 
note,  of  the  scale,  we  measure  off  the  following  fractions  of  the 
whole  for  the  successive  notes,  namely:  f,  |,  |,  •*,  f,  T85,  ^.    If  the 
whole,  and  these  fractions,  are  made  to  vibrate  in  order,  the  ear 
will  recognize  the  sounds  as  forming  the  series  called  the  gamut, 
or  diatonic  scale.    And  the  interval  between  the  fundamental  and 
each  of  the  others  is  named  according  to  its  distance  inclusively. 
Thus,  the  interval  from  the  whole  (=  1)  to  f,  is  called  the  second; 
from  1  to  |,  the  third,  &c. ;  therefore,  from  1  to  \,  the  eighth,  or 
octave.    Now,  as  the  number  of  vibrations  varies  inversely  as  the 
length  of  the  string,  the  numbers  corresponding  to  the  notes  re- 
spectively, are  expressed  by  the  same  fractions  inverted,  1,  f ,  f ,  |, 
-J,  |,  Jg5,  2.    Eeducing  these  to  a  common  denominator,  and  using 
the  numerators  (since  they  have  the  same  ratios),  we  have  the  fol- 
lowing series,  24,  27,  30,  32,  36,  40,  45,  48,  to  express  in  the  sim- 
plest manner  the  relative  numbers  of  vibrations  in  the  notes  of 
the  scale,  however  produced.    The  sounds  represented  by  these 
numbers  are  not  arbitrarily  chosen  to  form  the  scale,  but  they  are 
demanded  by  the  ear,  and  constitute  the  basis  of  the  music  of  all 
ages  and  nations. 

340.  Interval  of  the  Second. — In  examining  the  relation 
of  each  two  successive  numbers  in  the  foregoing  series,  we  find 
three  different  ratios.     Thus, 

27  :  24,  36  :  32,  and  45  :  40,  is  each  as    9  :    8. 

30  :  27 and  40  :  36, 10  :    9. 

32  :  30 and  48  :  45, 16  :  15. 

Therefore,  of  the  seven  intervals,  called  the  second,  in  the  dia- 
tonic scale,  there  are  three  equal  to  f ,  two  equal  1y°,  and  two  others 
equal  to  \%.  Each  of  the  first  five  is  called  a  major  second,  or  a 
tone  ;  each  of  the  last  two  is  called  a  minor  second,  or  a  semitone. 


THE    DIATONIC    SCALE.  215 

One  of  the  larger  tones  exceeds  one  of  the  smaller  by  f  J ;  for 
|  -T-  Jy°  =  f  J.  This  small  interval,  f  J,  is  called  a  comma,  and  is 
employed  as  a  measuring  unit  in  estimating  the  relations  of  inter- 
vals. The  minor  second,  though  called  a  semitone,  is  in  fact 
more  than  half  of  either  kind  of  major  second. 

341.  Repetition  of  the  Scale.— The  eighth  note  of  the 
scale  so  much  resembles  the  first  in  sound,  that  it  is  regarded  as  a 
repetition  of  it,  and  called  by  the  same  name.     Beginning,  there- 
fore, with  the  half  string,  where  the  former  series  closed,  let  us 
consider  the  sound  of  that  as  the  fundamental,  and  take  f  of  it 
for  the  second,  f  of  it  for  the  third,  &c. ;  we  then  close  a  second 
series  of  notes  on  the  quarter-string,  whose  sound  is  also  consid- 
ered a  repetition  of  the  former  fundamental.     Each  fraction  of 
the  string  used  in  the  second  scale  is  obviously  half  of  the  corre- 
sponding fraction  of  the  whole  string,  and  therefore  its  note  an 
octave  above  the  note  of  that.     This  process  may  be  repeated  in- 
definitely, giving  the  second  octave,   third  octave,  &c.    Ten  or 
eleven  octaves  comprehend  all  sounds  appreciable  by  the  human 
ear ;  the  vibrations  of  the  extreme  notes  of  this  entire  range  have 
the  ratio  of  1  :  2'°,  or  1:2";  that  is,  1  :  1024,  or  1 :  2048.     Hence, 
if  16  vibrations  per  second  produce  the  lowest  appreciable  note, 
the  highest  varies  from  16,000  to  33,000.    It  was  ascertained  by 
Dr.  Wollaston  that  the  highest  limit  is  different  for  different  ears ; 
so  that  when  one  person  complains  of  the  piercing  shrillness  of  a 
sound,  another  maintains  that  there  is  no  sound  at  all.    The 
lowest  limit  is  indefinite  for  a  different  reason ;  the  sounds  are 
heard  by  all,  but  some  will  recognize  them  as  low  musical  tones, 
while  others  only  perceive  a  rattling  or  fluttering  noise.    Few  mu- 
sical instruments  comprehend  more  than  six  octaves,  and  the 
human  voice  has  only  from  one  to  three,  the  male  voice  being  in 
pitch  an  octave  lower  than  the  female. 

342.  Modes  of  Naming  the  Notes. — There  is  one  system 

of  names  for  the  notes  of  the  scale,  which  is  fixed,  and  another 
which  is  movable.  The  first  is  by  the  seven  letters,  A,  B,  C,  D,  E, 
F,  G.  The  notes  of  the  second  octave  are  expressed  by  the  same 
letters,  in  some  way  distinguished  from  the  former. '  The  best 
method  is  to  write  by  the  side  of  the  letter  the  numeral  expressing 
that  index  of  2,  which  corresponds  to  the  octave:  as  A  ,  A^  &c., 
in  the  octaves  above;  A-,  A^  in  those  below. 

The  second  mode  of  designation  is  by  the  syllables,  do,  re,  mi, 
fa,  sol,  la,  si.  These  express  merely  the  relations  of  notes  to  each 
other,  do  always  being  the  fundamental,  re  its  second,  mi  its  third, 
&c.  In  the  natural  scale,  do  is  on  the  letter  C,  re  on  D,  &c. ;  but 
by  the  aid  of  interpolated  notes,  the  scale  of  syllables  may  be 


216  SOUND. 

transferred,  so  as  to  begin  successively  with  every  letter  of  the 
fixed  scale. 

343.  The  Chromatic  Scale. — Let  the  notes  of  the  diatonic 
scale  be  represented  (Fig.  207)  by  the  horizontal  lines,  C,  D,  &c. ; 
the  distance  from  C  to  D  being  a  tone,  from  D  to  E  a 

tone,  E  to  F  a  semitone,  &c.  It  will  be  observed  that  FlG  207> 
the  fundamental,  C,  is  so  situated  that  there  are  two  -0 
whole  tones  above  it,  before  a  semitone  occurs,  and  then 
three  whole  tones  before  the  next  semitone.  C  is  there-  c 
fore  the  letter  to  be  called  by  the  syllable  do,  in  order  to  ja 
bring  the  first  semitone  between  the  3d  and  4th,  and 
the  other  semitone  between  the  7th  and  8th,  as  the  ^ 
figure  represents  them.  Now,  that  we  may  be  able  to 
transfer  the  scale  of  relations  to  every  part  of  the  fixed  ^ 
scale  (which  is  necessary,  in  order  to  vary  the  character 
of  music,  without  throwing  it  beyond  the  reach  of  the 
voice),  the  whole  tones  are  bisected,  and  two  semitone 
intervals  occupy  the  place  of  each.  The  dotted  lines  in  ^ 
the  figure  show  the  places  of  the  interpolated  notes, 
which,  with  the  original  notes  of  the  diatonic  scale,  di-  -° 
vide  the  whole  into  a  series  of  semitones.  This  is  called 
the  chromatic  scale.  The  interpolated  note  between  G  C 
and  D  is  written  C$  (C  sharp),  or  D\y  (D  flat),  and  so  JB- 
of  the  others.  As  the  whole  tones  lie  in  groups  of  tivos 
and  threes,  so  the  new  notes  inserted  are  grouped  in  the 
same  way.  This  explains  the  arrangement  of  the  Uack 
keys  by  twos  and  threes  alternately  in  the  key-board  of  the  organ 
and  piano-forte.  The  white  keys  compose  the  diatonic  scale,  the 
white  and  black  keys  together,  the  chromatic  scale.  It  is  obvious 
that  on  the  chromatic  scale  any  one  of  the  twelve  notes  which 
compose  it  may  become  do,  or  the  fundamental  note,  since  the  re- 
quired series,  2  tones,  1  semitone,  3  tones,  1  semitone,  can  be  ar- 
ranged to  succeed  each  other,  at  whatever  note  we  begin  the 
reckoning.  This  change,  by  which  the  fundamental  note  is  made 
to  fall  on  different  letters,  is  called  the  transposition  of  the  scale. 

344.  Chords  and  Discords. — When  two  or  more  sounds, 
meeting  the  ear  at  once,  form  a  combination  which  is  agreeable, 
it  is  called  a  chord;  if  disagreeable,  a  discord.     The  disagreeable 
quality  of  a  discord,  if  attended  to,  will  be  perceived  to  consist  in 
a  certain  roughness,  or  harshness,  however  smooth  and  pure  the 
simple  sounds  which  are  combined.     On  examining  the  combina- 
tions, it  will  be  found  that  if  the  vibrations  of  two  sounds  are  in 
some  very  simple  relations,  as  1  :  2,  1  :  3,  2  :  3,  3  :  4,  &c.,  they  pro- 
duce a  chord ;  and  the  lower  the  terms  of  the  ratio,  the  more  per- 


TEMPERAMENT.  217 

feet  the  chord.  On  the  other  hand,  if  the  numbers  necessary  to 
express  the  relations  of  the  sounds  are  large,  as  8  :  9,  or  15  : 16,  a 
discord  is  produced.  It  appears  that  concordant  sounds  have/re- 
qucnt  coincidences  of  vibrations.  If,  in  two  sounds,  there  is  coin- 
cidence at  every  vibration  of  each,  then  the  pitch  is  the  same,  and 
the  combination  is  called  unison.  If  every  vibration  of  one  coin- 
cides with  every  alternate  vibration  of  the  other,  the  ratio  is  1  :  2, 
and  the  chord  is  the  octave,  the  most  perfect  possible.  The  fifth  is 
the  next  most  perfect  chord,  where  every  second  vibration  of  the 
lower  meets  every  third  of  the  higher,  2  :  3.  The  fourth,  3  :  4,  the 
major  third,  4  :  5,  the  minor  third,  6  :  6,  and  the  sixth,  3  :  5,  are 
reckoned  among  chords ;  while  the  second,  8  :  9,  and  the  seventh, 
8  :  15,  are  harsh  discords.  What  is  called  the  common  chord  con- 
sists of  the  1st,  3d,  and  5th,  combined,  and  is  far  more  used  in 
music  than  any  other.  Harmony  consists  of  a  succession  of  chords, 
or  rather,  of  such  a  succession  of  combined  sounds  as  is  pleasing 
to  the  ear;  for  discords  are  employed  in  musical  composition, 
their  use  being  limited  by  special  rules.  Many  combinations, 
which  would  be  too  disagreeable  for  the  ear  to  dwell  upon,  or  to 
finish  a  musical  period,  are  yet  quite  necessary  to  produce  the  best 
effect ;  and  without  the  relief  which  they  give,  perfect  harmony, 
if  long  continued,  would  satiate. 

345.  Temperament. — This  is  a  term  applied  to  the  small 
errors  introduced  into  the  notes,  in  tuning  an  instrument  of  fixed 
keys,  in  order  to  adapt  the  notes  equally  to  the  several  scales.  If 
the  tones  were  all  equal,  and  if  semitones  were  truly  half  tones,  no 
such  adjustment  of  notes  would  be  needed;  they  would  all  be  ex- 
actly correct  for  every  scale.  Representing  the  notes  in  the  scale 
whose  fundamental  is  C  by  the  numbers  in  Art.  339,  we  have, 

0,  D,  E,  F,  G,   A,  B,  C.2,  D2,  E,,  &c. 
24,  27,  30,  32,  36,  40,  45,  48,  54,   60,  &c. 

Now  suppose  we  wish  to  make  D,  instead  of  C,  our  key-note ; 
then  it  is  obvious  that  E  will  not  be  exactly  correct  for  the  second 
on  the  new  scale.  For  the  fundamental  to  its  second  is  as  8  to  9 ; 
and  8  :  9  : :  27  :  30.375,  instead  of  30.  Therefore,  if  D  is  the  key- 
note, we  must  have  a  new  E,  slightly  above  the  E  of  the  original 
scale.  So  we  find  that  A,  represented  by  40,  will  not  serve  to  be 
the  5th  in  the  new  scale ;  since  2  :  3  :  :  27  :  40.5,  which  is  a  little 
higher  than  A  ( •—  40).  After  adding  these  and  other  new  notes, 
to  render  the  intervals  all  exactly  right  for  the  new  key  of  D,  if 
we  proceed  in  the  same  manner,  and  make  E  (—  30)  our  key-note, 
and  obtain  its  second,  third,  &c.,  exactly,  we  shall  find  some  of 
them  differing  a  little,  both  from  those  of  the  key  of  C,  and  also 


218  SOUND. 

of  the  key  of  D.  Using  in  this  way  all  the  twelve  notes  of  the 
chromatic  scale  in  succession  for  the  fundamental,  it  appears  that 
several  different  E's,  F's,  Gr's,  &c.,  are  required,  in  order  to  make 
each  scale  perfect.  In  instruments,  whose  sounds  cannot  be  mod- 
ified by  the  performer,  like  the  organ  and  piano-forte,  as  it  is  con- 
sidered impossible  to  insert  all  the  pipes  or  strings  necessary  to 
render  every  scale  perfect,  such  an  adjustment  is  made  as  to  dis- 
tribute these  errors  equally  among  all  the  scales.  For  example,  E 
is  not  made  a  perfect  third  for  the  key  of  '(7,  lest  it  should  be  too 
imperfect  for  a  second  in  the  key  of  D,  and  for  its  appropriate 
place  in  other  scales.  It  is  this  equal  distribution  of  errors  among 
the  several  scales  which  is  called  temperament.  The  errors,  when 
thus  distributed,  are  too  small  to  be  observed  by  most  persons ; 
whereas,  if  an  instrument  was  tuned  perfectly  for  any  one  scale, 
all  others  would  be  intolerable. 

The  word  temperament,  as  above  explained,  has  no  application 
except  to  instruments  of  fixed  keys,  as  the  organ  and  piano-forte ; 
for,  where  the  performer  can  control  and  modify  the  notes  as  he  is 
playing,  he  can  make  every  key  perfect,  and  then  there  are  no 
errors  to  be  distributed.  The  flute-player  can  roll  the  flute 
slightly,  and  thus  humor  the  sound,  so  as  to  cause  the  same 
fingering  to  give  a  precisely  correct  second  for  one  scale,  a  correct 
third  for  another,  and  so  on.  The  player  on  the  violin  does  the 
same,  by  touching  the  string  in  points  slightly  different.  The  or- 
gans of  the  voice,  especially,  can  be  adjusted  to  make  the  intervals 
perfect  on  every  scale.  In  these  cases  there  is  no  tempering,  or 
dividing  of  errors  among  different  scales,  but  a  perfect  adjustment 
to  each  scale,  by  which  all  error  is  avoided. 

346.  Harmonics. — The  fact  has  been  mentioned  that  a  string, 
or  a  column  of  air,  may  vibrate  in  parts,  even  while  vibrating  as  a 
whole.  It  only  remains  to  show  the  musical  relations  of  the 
sounds  thus  produced.  When  a  string  vibrates  in  parts,  it  divides 
into  halves,  thirds,  fourths,  or  other  aliquot  parts.  Now,  a  half- 
string  produces  an  octave  above  the  whole,  making  the  most  per- 
fect chord  with  it.  The  third  of  a  string  being  two-thirds  of  the 
half-string,  produces  the  fifth  above  the  octave,  a  very  perfect 
chord.  The  quarter-string  gives  the  second  octave  ;  the  fifth  part 
of  it,  being  |  of  the  quarter,  gives  the  major  third  above  the 
second  octave ;  and  the  sixth  part,  being  f  of  the  quarter,  gives 
the  fifth  above  the  second  octave.  Thus,  all  the  simpler  divisions, 
which  are  the  ones  most  likely  to  occur,  are  such  as  produce  the 
best  chords ;  and  it  is  for  this  reason  that  the  sounds  are  called 
harmonics.  The  same  is  true  of  air-columns  and  bells.  The 
-ZEolian  harp  furnishes  a  beautiful  example  of  the  harmonics  of  a 


OVERTONES.  219 

string.  Two  or  more  fine  smooth  cords  are  fastened  upon  a  box, 
and  tuned,  at  suitable  intervals,  like  the  strings  of  a  violin ;  and 
the  box  is  placed  in  a  narrow  opening,  where  a  current  of  air 
passes.  Each  string  at  different  times,  according  to  the  intensity 
of  the  breeze,  will  emit  a  pure  musical  note ;  and,  with  every 
change,  will  divide  itself  in  a  new  mode,  and  give  another  pitch, 
while  it  will  frequently  happen  that  the  vibrations  of  different 
divisions  will  coexist,  and  their  harmonic  sounds  mingle  with  each 
other. 

347.  Overtones. — But  the  parts  into  which  a  sounding  body 
divides  do  not  always  harmonize  with  the  whole.    For  instance,  J- 
or  TV  of  a  string  is  discordant  with  the  fundamental.    The  word 
harmonics  is  not,  therefore,  applicable  except  to  a  very  few  of  the 
many  possible  sounds  which  a  body  may  produce.    The  word 
overtone  is  used  to  express  in  general  any  sound  whatever,  given 
by  a  part  of  a  sounding/  body.     A  string  may  furnish  20  or  30 
overtones,  but  only  a  small  number  of  them  would  be  harmonics. 

348.  Quality  of  Tone. — Even  when  the  pitch  of  two  sound- 
ing bodies  is  the  same,  the  ear  almost  always  distinguishes  one 
sound  from  the  other  by  certain  qualities  of  tone  peculiar  to  each. 
Thus,  if  the  same  letter  be  sounded  by  a  flute  and  the  string  of  a 
piano,  each  note  is  easily  distinguished  from  the  other.     TWTO 
church-bells  may  be  upon  the  same  key,  and  yet  one  be  agreeable, 
and  the  other  harsh  to  the  ear.     While  these  great  diversities  may 
to  some  extent  be  due  to  circumstances  not  yet  discovered,  still  it 
is  certain  that  they  in  no  small  degree  arise  from  the  vibrations 
of  various  parts  mingling  with  those  of  the  fundamental  sound. 
A  long  monochord  can,  by  varying  the  mode  of  exciting  the  vibra- 
tions, be  made  to  yield  a  great  variety  of  sounds,  while  there  is 
perceived  in  them  all  the  same  fundamental  undertone  which  de- 
termines the  pitch.    If  the  string  be  struck  at  the  middle,  then  no 
node  can  be  formed  at  that  point ;  hence,  the  mixed  sound  will 
contain  no  overtones  of  the  J,  \,  i,  |,  TL,  or  other  even  aliquot 
parts  of  the  string ;  for  all  such  would  require  a  node  at  the  mid- 
dle.   But  if  struck  at  one-third  of  its  length  from  the  end,  then 
the  overtones,  12>  -}>  &c->  maJ  exist,  but  not  those  of  |,  J,  J,  or  any 
other  parts  whose  node  would  fall  at  |  of  the  length  from  the 
end. 

For  reasons  which  are  mostly  unknown,  some  sounding  bodies 
have  their  fundamental  accompanied  by  harmonic  overtones,  and 
others  by  overtones  which  are  discordant.  And  this  is  one  cause 
of  the  agreeable  or  unpleasant  quality  of  the  sounds  of  different 
bodies. 


220  SOUND. 

349.  Communication  of  Vibrations. — The  acoustic  vibra- 
tions of  one  body  are  readily  communicated  to  others,  which  are 
near  or  in  contact.    We  have  already  noticed  that  the  vibrations 
of  a  reed  will  excite  those  of  a  column  of  air  in  a  pipe.    If  two 
strings,  which  are  adapted  to  vibrate  alike,  are  fastened  on  the 
same  box,  and  one  of  them  is  made  to  sound,  the  other  will  sound 
also  more  or  less  loudly,  according  to  the  intimacy  of  their  connec- 
tion.   The  vibrations  are  communicated  partly  through  the  air, 
and  partly  through  the  materials  of  the  box.    So,  if  a  loud  sound 
is  uttered  near  a  piano-forte,  several  strings  will  be  thrown  into 
vibration,  whose  notes  are  heard  after  the  voice  ceases.    The  no- 
ticeable fact  in  all  sdch  experiments  is,  that  the  vibrations  thus 
communicated  from  one  body  to  another  cause  sounds  which  har- 
monize with  each  other,  and  with  the  original  sound.    For  the 
rate  of  vibration  will  either  be  identical,  or  have  those  simple  re- 
lations which  are  expressed  by  the  smallest  numbers.    Let  a  per- 
son hold  a  pneumatic  receiver  or  a  large  tumbler  before  him,  and 
utter  at  the  mouth  of  it  several  sounds  of  different  pitch ;  and.  he 
will  probably  find  some  one  pitch  which  will  be  distinctly  rein- 
forced by  the  vessel.    That  particular  note,  which  the  receiver  by 
its  size  and  form  is  adapted  to  produce,  will  not  be  called  forth  by 
a  sound  that  would  be  discordant  with  it.    The  melodeon,  ser- 
aphine,  and  instruments  of  like  character,  owe  their  full  and  bril- 
liant notes  to  reeds,  each  of  which  has  its  cavity  of  air  adapted  to 
vibrate  in  unison  with  it.    It  sometimes  happens  that  the  second 
body,  vibrating  as  a  whole,  would  not  harmonize  with  the  first,  and 
yet  will  give  the  same  note  by  some  mode  of  division.    Thus  it  is 
that  all  the  various  sounds  of  the  monochord,  and  of  the  strings 
of  the  viol,  are  reinforced  by  the  case  of  thin  wood  upon  which 
they  are  stretched.     The  plates  of  wood  divide  by  nodal  lines  into 
some  new  arrangement  of  ventral  segments  for  every  new  sound 
emitted  by  the  string.    In  like  manner,  the  pitch  of  the  tuning- 
fork,  and  all  the  rapid  notes  of  a  music-box,  are  rendered  loud  and 
full  by  the  table,  in  contact  with  which  they  are  brought.    The 
extended  material  of  the  table  is  capable  of  division  into  a  great 
variety  of  forms,  and  will  always  give  a  sound  in  unison  with  the 
instrument  which  touches  it. 

350.  One  System  of  Vibrations  Controlling  Another.— 

If  two  sounding  bodies  are  nearly,  but  not  precisely  on  the  same 
key,  they  will  sometimes,  when  brought  into  close  contact,  be 
made  to  harmonize  perfectly.  The  vibrations  of  the  more  power- 
ful will  be  communicated  to  the  other,  and  control  its  movements 
so  that  the  discordance,  which  they  produce  when  a  few  inches 
apart,  will  cease,  and  concord  will  ensue.  Two  diapason  pipes  of 


CRISPATIONS    OF    FLUIDS.  221 

an  organ,  tuned  a  quarter-tone  or  even  a  semitone  from  unison,  so 
as  to  jar  disagreeably  upon  the  ear,  when  one  inch  or  more  asun- 
der, will  be  in  perfect  unison,  if  they  are  in  contact  through  their 
whole  length.  Even  the  slow  oscillations  of  two  watches  will  in- 
fluence each  other ;  if  one  gains  on  the  other  only  a  few  beats  in 
an  hour,  then,  if  they  arc  placed  side  by'side  on  the  same  board, 
they  will  beat  precisely  together. 

351.  Crispations  of  Fluids. — Among  the  numerous  acous- 
tic experiments  illustrating  the  communication  of  vibrations,  none 
are  more  beautiful  than  those  in  which  the  vibrations  of  glass  rods 
are  conveyed  to  the  surface  of  a  fluid.     Let  a  very  shallow  pan  of 
glass  or  metal  be  attached  to  the  middle  of  a  thin  bar  of  wood, 
three  or  four  feet  long,  and  resting  near  its  ends  on  two  fixed 
bridges ;  let  water  be  placed  in  the  pan,  and  a  long  glass  rod 
standing  in  it,  or  on  the  wood,  be  vibrated  longitudinally,  by 
drawing  the  moistened  fingers  down  upon  it ;  the  liquid  immedi- 
ately shows  that  the  vibrations  are  communicated  to  it.    The  sur- 
face is  covered  with  a  regular  arrangement  of  heaps,  called  crispar 
tions,  which  vary  in  size  with  the  pitch  of  sound,  which  is  produced 
by  the  same  vibration.    If  the  pitch  is  higher,  they  are  smaller, 
and  may  be  readily  varied  from  three  or  four  inches  in  diameter  to 
the  fineness  of  the  teeth  of  a  file.     Crispations  of  the  same  charac- 
ter are  also  formed  in  clusters  on  the  water  in  a  large  tumbler  or 
glass  receiver,  when  the  finger  is  drawn  along  its  edge ;  every  ven- 
tral segment  of  the  glass  produces  a  group  of  hillocks  by  the  side 
of  it  on  the  surface  of  the  water. 

352.  Interference  of  Waves  of  Sound. — Whenever  two 
sounds  are  moving  through  the  air,  every  particle  will,  at  a  given 
instant,  have  a  motion  which  is  the  resultant  of  the  two  motions 
which  it  would  have  had  if  the  sounds  were  separate.    These  mo- 
tions may  conspire,  or  they  may  oppose  each  other.    The  word  in- 
terference  is  used  in  scientific  language  to  express  the  resultant 
effect,  whatever  it  may  be.    The  beats,  which  are  frequently  heard 
in  listening  to  two  sounds,  indicate  the  points  of  maximum  con- 
densation produced  by  the  union  of  the  condensed  parts  of  both 
systems  of  waves.    And  the  sounds  are  considered  discordant  when 
these  beats  are  just  so  frequent  as  to  produce  a  disagreeable  flut- 
tering or  rattling.     If  too  near  or  too  far  apart  for  this,  they  are 
regarded  practically  as  concordant.    And  when  the  beats  are  too 
close  to  be  perceived  separately,  yet  the  peculiar  adjustment  of 
condensations  of  one  system  with  those  of  the  other,  according  as 
one  wave  measures  two,  or  two  waves  measure  three,  or  four 
measure  five,  &c.,  is  at  once  distinguished  by  the  ear,  and  recog- 
nized as  the  chord  of  the  octave,  the  fifth,  the  third,  &c.    When  a 


222  SOUND. 

sound  and  its  octave  are  advancing  together,  there  are  instants  in 
which  any  given  particle  of  air  is  impressed  with  two  opposite  mo- 
tions, and  other  alternate  moments  when  both  motions  are  in  the 
same  direction.  For  the  waves  of  the  highest  sound  are  half  as 
long  as  those  of  the  lowest ;  hence,  while  every  second  condensa- 
tion of  the  former  coincides  with  every  condensation  of  the  latter, 
the  alternate  ones  of  the  former  must  be  at  the  points  of  greatest 
rarefaction  of  the  latter ;  and  this  cannot  occur  without  opposite 
movements  of  the  particles.  If  two  simultaneous  sounds  have  the 
same  pitch,  i.  e.,  the  same  length  of  wave,  they  ordinarily  run  to- 
gether, so  that  like  phases  in  the  two  systems  are  coincident,  and 
the  compound  sound  (called  unison)  simply  has  twice  the  loiidness 
of  one  of  them  alone.  But,  by  a  delicate  mode  of  experiment,  one 
of  these  two  systems  of  waves,  having  equal  lengths,  and  equal  in- 
tensities, may  be  made  to  fall  half  a  wave  behind  the  other,  in 
which  case  opposite  phases  coincide,  and  the  two  sounds  destroy 
each  other.  Thus,  two  sounds  produce  silence,  on  the  same  prin- 
ciple that  two  systems  of  water-waves  may  produce  level  water. 

353.  Number  and  Length  of  Waves  for  Each  Note.— 

Though  the  vibrations  of  any  musical  note  are  too  rapid  to  be 
counted,  yet  the  number  may  be  ascertained  in  several  ways.  One 
of  the  readiest  methods  is  by  means  of  a  little  instrument  called 
the  siren,  invented  by  De  La  Tour.  The  pulses  are  produced  by 
streams  of  air  driven  through  holes  in  a  revolving  wheel.  The 
revolutions  of  the  wheel  are  recorded  by  machinery,  and  the  num- 
ber of  vibrations  in  each  revolution  is  known  from  the  number  of 
holes  through  which  the  air  rushes.  When  such  a  velocity  of 
revolution  is  given  as  to  produce  the  required  pitch,  then  the  rev- 
olutions of  the  index  per  minute  may  be  counted,  and  the  number 
of  vibrations  in  the  same  time  will  be  known,  and  therefore  the 
number  per  second.  In  this  and  other  ways  it  is  ascertained  that 
the  numbers  corresponding  to  the  letters  of  the  scale  are  the 
following : 

(   O,     D,     E,    F,      G,     A,      B,     C2,  D,, 
(128,  144, 160,  170§,  192,  213|,  240,  256,  288. 

The  highest  note  of  the  above  series,  D.2,  288,  is  the  lowest  on  the 
common  or  Z)-flute.  There  is  not,  however,  a  perfect  agreement 
of  pitch  in  different  countries,  and  among  different  classes  of  mu- 
sicians. Accordingly,  C,  which  is  given  above  as  corresponding  to 
128  vibrations  per  second,  has  several  values,  varying  from  127  to 
131. 

To  find  the  length  of  acoustic  waves  for  any  given  pitch,  we 
have  only  to  divide  the  velocity  of  sound  in  one  second  by  the 


VIBRATIONS    VISIBLY    PROJECTED.  223 

number  of  vibrations  which  reach  the  ear  in  the  same  length  of 
time.  For  example,  at  the  temperature  of  60°,  sound  travels  1118 
feet  per  second ;  therefore  the  length  of  waves  of  low  D  on  the 
flute  =  1118  -i-  288  =  nearly  four  feet.  The  waves  of  the  lowest 
musical  note  are  about  70  feet  long ;  and  of  the  highest,  less  than 
half  an  inch. 

354.  Acoustic  Vibrations  Visibly  Projected.— The  vi- 
brations of  heavy  tuning-forks  can  be  magnified  and  rendered  dis- 
tinctly visible  to  an  audience  by  projecting  them  on  a  screen. 
The  fork  being  constructed  with  a  small  metallic  mirror  attached 
near  the  end  of  one  prong,  a  sunbeam  reflected  from  the  mirror 
will  exhibit  all  the  movements  of  the  fork  greatly  enlarged  on  a 
distant  wall ;  and  if  the  fork  is  turned  on  its  axis,  the  luminous 
projection  will  take  the  form  of  a  waving  line.  And  by  the  use 
of  two  forks,  all  the  phenomena  of  interference  may  be  rendered 
as  distinct  to  the  eye  as  they  are  to  the  ear. 


PART  Y. 

M-A-GISrETISM 


CHAPTER  I. 

THE  MAGNET  AND  ITS  PROPERTIES. 

355.  The  Magnet. — Fragments  of  iron  ore  are  sometimes 
found  which  strongly  attract  iron ;  and  bars  of  steel  are  artificially 
prepared  which  exhibit  the   same  property.      These  bodies  are 
called  magnets ;  the  ore  is  the  natural  magnet,  commonly  called 
lodestone ;  the  prepared  steel  bar  is  an  artificial  magnet. 

Another  property  distinguishes  the  magnet,  namely,  that  when 
properly  suspended  on  a  pivot,  it  assumes  a  certain  definite  direc- 
tion with  regard  to  the  earth.  This  property  of  the  magnet  is 
called  its  polarity. 

356.  The  Attraction  Between  a  Magnet   and   Iron.— 

The  magnetic  property  which  is  likely  to  be  first  noticed  is  the 
attraction  of  iron.  If  a  lodestone  or  a  bar  magnet  be  rolled  in 
iron  filings  (Fig.  208),  there  are  two  opposite  points  to  which  the 

FIG.  208. 


filings  attach  themselves  in  thick  clusters,  arranged  in  diverging 
filaments.  These  opposite  points  of  greatest  action  are  called 
poles.  The  attraction  diminishes  from  the  poles  towards  the  cen- 
tral parts ;  and  about  in  the  middle  between  them  there  is  little 
or  none;  this  is  called  the  neutral  point.  The  straight  line  from 
one  pole  to  the  other  is  called  the  axis. 

The  mutual  attraction  between  a  magnet  and  iron  is  shown 
by  bringing  a  piece  of  iron  toward  either  pole  of  the  magnetic 
needle ;  the  needle  instantly  turns  so  as  to  bring  its  pole  as  near 


POLARITY. 


On  the  other  hand,  an  iron 


FIG.  209. 


as  possible  to  the  iron  (Fig.  209). 
needle  being  suspended  in 
like  manner,  the  same  move- 
ment takes  place,  when 
either  pole  of  a  magnet  is 
brought  near  to  it. 

Nickel,  cobalt,  and  some- 
times manganese,  exhibit 
the  same  magnetic  proper- 
ties in  some  degree.  But 
these  exist  in  comparatively 
small  quantities,  and  therefore  by  magnetic  bodies  are  usually  in- 
tended only  iron  and  steel. 


FIG.  210. 


357.  Polarity. — If  a  light  magnet  is  delicately  suspended  on 
a  pivot  at  the  neutral  point,  as  in  Fig.  210,  it  is  called  a  magnetic 
needle.    When  thus  placed  and  left  to  it- 
self, it  oscillates  for  a  time,  and  finally 

settles  with  its  axis  in  a  certain  fixed  di- 
rection, which  in  most  places  is  nearly 
north  and  south.  The  end  which  points 
in  a  northerly  direction  is  called  the  north 
pole;  the  other,  the  south  pole.  These 
poles  are  usually  marked  on  the  larger 
magnets  by  the  letters  N  and  8,  so  that  they  may  be  instantly 
distinguished.  If  a  magnetic  needle  has  simply  a  mark  or  stain 
on  one  end,  that  end  is  understood  to  be  the  north  pole. 

358.  Action  of  Magnets  on  Each  Other. — While  either 
pole  of  a  magnet  attracts  and  is  attracted  by  a  piece  of  iron,  it  is 
otherwise  when  the  pole  of  one  magnet  is  brought  near  the  pole 
of  another.    There  is  attraction  in  some  cases,  and  repulsion  in 
others.    If  the  magnets  are  properly  marked,  and  one  of  them 
suspended  so  as  to  move  freely,  it  is  readily  discovered  that  tlio 
law  of  action  is  the  following : 

Poles  of  the  same  name  repel,  and 
those  of  contrary  name  attract  each 
other. 

Thus,  the  pole  8  of  the  magnet 
(Fig.  211)  repels  s  of  the  needle,  and 
attracts  n ;  and  if  the  magnet  were 
inverted,  and  the  pole  N  brought 
near  to  n,  the  latter  would  be  re- 
pelled, and  s  be  attracted. 
15 


FIG.  211. 


226  MAGNETISM. 

359.  Magnetic  Induction. — When  a  bar  of  iron  is  brought 
near  to  the  pole  of  a  magnet,  though  attraction  is  the  phenome- 
non first  observed,  as  stated  (Art.  356),  yet  it  is  readily  proved 
that  this  attraction  results  from  a  change  which  is  previously  pro- 
duced in  the  iron.    It  becomes  a  magnet  through  the  influence  of 
the  magnet  which  is  near  it.     That  end  of  the  iron  bar  which  is 
placed  near  one  pole  of  a  magnet  becomes  a  pole  of  the  opposite 
name,  and  the  remote  end  a  pole  of  the  same  name.    Hence,  ac- 
cording to  the  law  (Art.  358),  the  poles  which  are  contiguous  at- 
tract each  other,  because  they  are  unlike.    The  influence  by  which 
the  iron  becomes  a  magnet  is  called  induction.    A  magnet,  when 
brought  near  to  a  piece  of  iron,  induces  upon  the  iron  the  mag- 
netic condition,  without  any  loss  of  its  own  magnetic  properties. 
This  influence  is  more  powerful  according  as  the  two  are  nearer  to 
each  other ;  it  is,  therefore,  greatest  when  the  two  bars  are  in 
contact. 

That  the  iron  is  truly  a  magnet  for  the  time  being  is  proved 
by  bringing  a  needle  near  to  its  remote  end ;  one  pole  is  attracted, 
and  the  other  repelled.  If  the  iron  had  not  been  changed  into  a 
magnet,  each  pole  of  the  needle  would  be  attracted  by  it  (Art.  356). 

360.  Successive  Inductions, — Let  a  bar  of  iron,  B  (Fig. 
212),  be  suspended  from  the  south  pole  of  the  magnet  A  ;  then 
the  upper  end  of  B  is  a  north  pole,  and  the  lower  end  a  south  pole. 
Now,  as  B  is  a  magnet,  it  will  in- 
duce the  magnetic  state  on  another 

bar,  (7,  when  brought  in  contact; 
and,  as  before,  the  poles  of  opposite 
name  will  be  contiguous.  There- 
fore, the  upper  end  of  C  is  north, 
arid  the  lower  end  south.  D  is  also 
a  magnet  by  the  inductive  power 
of  C.  Thus,  there  is  an  indefinite  series  of  inductions,  growing 
•  weaker,  however,  from  one  to  another,  as  the  number  is  greater, 
and  as  the  bars  are  longer. 

The  filaments  of  iron  filings  which  attach  themselves  to  the 
pole  of  a  magnet  (Art.  356)  are  so  many  series  of  small  magnets 
formed  in  the  same  manner  as  just  described.  Every  particle  of 
iron  is  a  complete  magnet,  having  its  poles  so  arranged  that  the 
opposite  poles  of  two  successive  particles  are  always  contiguous. 

361.  Reflex  Influence.— When  a  magnet  exerts  the  induc- 
tive power  upon  a  piece  of  iron  which  is  near  it,  its  own  magnetic 
intensity  is  increased.    The  end  of  the  piece  of  iron  contiguous  to 
the  pole  of  the  magnet  is  no  sooner  endued  with  the  opposite  polar- 
ity than  it  reacts  upon  the  magnet,  and  increases  its  intensity  ;  so 


MAGNETIC  .INDUCTION.  227 

that,  if  fragments  of  iron  are  attached  to  a  magnet,  as  many  as  it 
will  sustain,  then  after  a  time  another  may  be  added,  and  again 
another,  till  there  is  a  very  sensible  increase  of  its  original 
power. 

Hence,  too,  the  force  of  attraction  of  the  dissimilar  poles  of 
two  magnets  is  greater  than  the  force  of  repulsion  of  the  similar 
poles ;  because,  when  the  poles  are  unlike,  each  acts  inductively 
on  the  other  to  develop  its  poles  more  fully ;  but  when  they  are 
alike,  the  influence  which  they  reciprocally  exert  tends  to  make 
them  unlike,  and  of  course  to  diminish  their  repulsive  force. 

An  extreme  case  of  this  diminution  of  repulsive  force  occurs 
when  the  like  poles  of  two  very  unequal  magnets  are  brought  into 
contact.  The  small  magnet  immediately  clings  to  the  large  one, 
as  though  the  poles  were  unlike ;  and  if  examined,  it  is  found  that 
they  are  unlike.  The  powerful  magnet  has  in  an  instant  reversed 
the  poles  of  the  weak  one  by  its  strong  inductive  power,  the  latter 
not  having  force  enough  to  diminish  sensibly  the  strength  of  the 
other. 

362.  Double  Induction. — The  effects  of  two  inductions  at 
once  on  a  bar  of  iron  are  various. 

1.  The  bar  may  become  a  single  magnet  of  double  strength. 

2.  It  may  consist  of  two  distinct  magnets. 

3.  It  may  have  no  magnetic  power  at  all. 

The  first  case  is  illustrated  by  bringing  the  north  pole  of  a 
magnet  to  one  end  of  the  iron,  and  the  south  pole  of  another  mag- 
net to  the  other  end.  Each  magnet  will  form  two  poles  by  induc- 
tion, and  it  is  evident  that  the  two  pairs  of  poles  will  coincide. 
Even  one  magnet  produces  the  same  effect  when  laid  by  the  side 
of  a  bar  of  iron  of  the  same  length. 

To  show  the  second  effect,  apply  one  pole  of  a  magnet  to  the 
middle  of  the  iron  bar ;  then  an  opposite  pole  is  formed  at  the 
middle,  and  a  like  pole  at  each  end,  each  half  of  the  bar  being  a 
separate  magnet.  The  same  effect  is  produced  by  bringing  the 
like  poles  of  two  magnets  in  contact  with  the  ends  of  the  bar ;  for 
both  ends  will  be  of  the  opposite  kind,  and  the  middle  of  the  same 
kind,  as  the  poles  applied.  If  a  pole  is  applied  to  the  middle  of  a 
star  of  iron,  the  extremity  of  each  ray  is  a  pole  of  the  same  kind ; 
if  to  the  middle  of  a  circle  of  iron,  the  same  polarity  is  found  a£ 
every  point  of  the  circumference. 

As  an  example  of  the  third  case,  suspend  a  bar  of  iron  from 
the  pole  of  a  magnet,  and  then  bring  the  opposite  pole  of  an  equal 
magnet  to  the  point  of  contact ;  the  two  poles  induced  by  one  are 
contrary  to  the  two  induced  by  the  other,  and  they  are  found  to 
be  completely  neutralized. 


228  MAGNETISM. 

This  last  case  shows  that  two  opposite  and  equal  magnetic 
poles  formed  at  the  same  point  destroy  each  other. 

363.  Coercive  Force. — If  in  the  several  experiments  on  iron 
bars,  which  have  been  already  described,  pure  annealed  iron  is 
used,  the  effects  take  place  instantly ;  and  when  the  magnet  is  re- 
moved, they  as  suddenly  disappear.    But  if  the  iron  is  hard,  mag- 
netic poles  are  developed  in  it  slowly ;  and  when  they  have  been 
developed,  the  iron  returns  also  slowly  to  its  neutral  condition. 

That  property  of  hard  unannealed  iron  which  obstructs  the 
development  of  magnetism  in  it,  and  which  hinders  its  return  to 
a  neutral  state,  is  called  the  coercive  force.  In  iron  which  is  pure 
and  well  annealed  there  seems  to  be  no  coercive  force.  It  appears 
in  a  slight  degree  in  iron  not  carefully  prepared,  and  increases 
with  its  hardness ;  it  is  great  in  tempered  steel,  which  is  a  com- 
pound of  iron  and  carbon,  and  greatest  of  all  in  steel,  which  is 
tempered  to  the  utmost  hardness. 

It  is,  therefore,  difficult  to  make  a  strong  magnet  of  a  steel  bar 
by  ordinary  induction,  unless  it  is  quite  thin ;  but  after  the  de- 
Telopment  has  once  been  made,  the  bar  becomes  a  permanent  mag- 
net, and  may  by  care  be  used  as  such  for  years. 

364.  Change  in  the  Coercive  Force.— The  coercive  force 
is  weakened  by  any  cause  which  excites  a  tremulous  or  vibratory 
motion  among  the  particles  of  the  steel.    This  happens  when  the 
bar  is  struck  by  a  hammer,  so  as  to  produce  a  ringing  sound, 
which  indicates  that  the  particles  are  thrown  into  a  vibratory  mo- 
tion.   The  passage  of  an  electric  discharge  through  a  steel  bar 
under  the  influence  of  a  magnet,  overcomes  the  coercive  force  for 
the  time  being,  and  permanent  magnetism  is  developed.    Heat 
produces  the  same  effect ;  and  hence  a  steel  bar  is  conveniently 
magnetized  by  heating  it  to  redness,  placing  it  under  a  powerful 
inductive  influence,  and  then  hardening  it  by  sudden  cooling. 
The  coercive  force  is  thus  neutralize'd  by  heat,  till  the  develop- 
ment takes  place,  when  it  is  restored,  and  the  bar  is  a  permanent 
magnet. 

A  magnet,  however,  loses  its  power  by  the  same  means  as,  dur- 
ing the  process  of  induction,  were  used  to  develop  it.  Accord- 
ingly, any  mechanical  concussion  or  rough  usage  impairs  or 
destroys  the  power  of  a  magnet.  By  falling  on  a  hard  floor,  or 
by  being  struck  with  a  hammer,  it  is  injured.  Heat  produces  a 
similar  effect.  A  boiling  heat  weakens,  and  a  red  heat  totally  de- 
stroys the  magnetism  of  a  needle. 

365.  Magnetism  not  Transferred,  but  only  Developed.— 

This  is  strikingly  proved  by  the  fact  that  if  a  magnet  be  divided, 


LAW    OF    FORCE    AND    DISTANCE.  229 

even  at  the  neutral  point,  where  there  is  no  sign  of  magnetism, 
the  parts  instantly  become  complete  magnets,  two  unlike  poles 
manifesting  themselves  at  the  place  of  fracture.  Both  polarities 
seem  to  exist  at  every  point,  and  are  developed  wherever  the  bar  is 
divided.  If  each  part  is  divided  again,  the  same  phenomenon  is 
repeated,  and  so  on  indefinitely.  There  is,  therefore,  no  transfer 
of  magnetism  from  one  point  to  another,  any  more  than  from  one 
bar  to  another,  but  only  an  excitation  of  what  existed  in  every 
part  of  the  body  before.  Both  the  north  and  the  south  pole  must 
be  conceived  as  latent  at  every  point  of  a  piece  of  iron  or  steel ; 
and  when  the  piece  is  magnetized,  either  north  or  south  polarity 
is  developed  more  or  less  fully  in  all  parts  except  the  neutral  point. 
It  is  not  necessary  that  the  particles  should  be  united  by  cohe- 
sion in  a  solid  bar.  A  magnet  can  be  formed  by  filling  a  brass 
tube  with  iron  filings  and  sand,  or  by  forming  a  rod  of  cement 
mixed  with  filings,  and  then  subjecting  them  to  inductive  influ- 
ence. Fig.  213  will  give  an 

idea  of  the  probable  struc-  _FlG'  213t 

ture  of  every  magnet.  Each 
particle  of  it  is  a  complete 
magnet,  the  like  poles  of  all 

are  turned  the  same  way,  and  unlike  poles  are  therefore  contigu- 
ous to  each  other,  and  each  acts  inductively  on  the  next. 

366.  Magnetic  Intensity  and  Distance.— The  law  of  the 

magnetic  force  is  the  following : 

The  intensity  of  the  magnetic  force,  whether  attraction  or  re- 
pulsion, varies  inversely  as  the  square  of  the  distance. 

The  law  in  the  case  of  the  repulsion  of  like  poles  is  readily 
proved  by  Coulomb's  torsion  balance,  which  is  figured  and  de- 
scribed in  Art.  402,  under  Electricity.  The  angle  of  torsion  is 
used  as  a  measure  of  the  repulsion,  and  it  is  found  that  the  wire 
must  be  twisted  through  four  times  as  large  an  angle  to  bring  the 
poles  to  one-half  the  distance,  and  nine  times  as  large  an  angle  to 
bring  them  to  one-third  the  distance,  &c.,  the  force  increasing  as 
the  square  of  the  distance  diminishes. 

To  prove  the  law  for  the  attraction  of  opposite  poles,  the  vibra- 
tions of  a  needle  are  counted,  when  it  is  placed  at  different  dis- 
tances from  a  magnet.  The  square  of  the  number  made  in  a  given 
time  is  a  measure  of  the  attractive  force,  just  as  the  square  of  the 
number  of  vibrations  of  a  pendulum  is  a  measure  of  the  force  of 
gravity  (Art.  170). 

In  each  of  these  experiments,  the  magnetic  influence  of  the 
earth  upon  the  needle  must  be  eliminated,  in  order  to  obtain  a 
correct  result. 


230 


MAGNETISM. 


367.   Equilibrium  of  a  Needle  Near  a  Maasiet— If  a 

small  needle,  free  to  revolve,  be  placed  near  the  pole  of  a  magnet, 
so  that  its  centre  is  in  the  axis  of  the  magnet  produced,  it  will 
place  itself  in  the  line  of  that  axis.     For  suppose  that  N  8  (Fig. 
214)  is  a  large  magnetic 
bar,  and  n  s  a  small  needle  ^IG> 

suspended  near  the  north 
pole  of  the  magnet,  with 
its  centre  in  the  axis  of 
the  bar  produced  at  a ;  it 
will  be  seen  that  the  ac- 
tion of  the  pole  of  the 
magnet  is  such  as  to  bring 
the  needle  into  a  line  with 
the  magnet.  The  action  of  the  pole  N  upon  the  needle  tending 
to  give  it  this  direction  (since  it  repels  n  and  attracts  s),  is  equal 
to  the  sum  of  its  actions  upon  both  poles.  The  pole  S,  by  repel- 
ling s,  and  attracting  n,  tends  to  reverse  this  position,  but,  on  ac- 
count of  greater  distance,  its  force  is  less  than  that  of  N. 

If  the  centre  of  the  needle  is  in  a  line  perpendicular  to  the  bar 
at  its  middle  point,  the  needle  will  be  in  equilibrium  when  paral- 
lel to  the  bar  with  its  poles  in  contrary  order.  Thus,  supposing 
the  needle  to  be  suspended  at  #,  it  will  be  seen  that  the  actions  of 
both  poles  of  the  magnet  conspire  to  move  n  to  the  left,  and  s  to 
the  right;  and  as  these  forces  are  equal,  equilibrium  takes  place 
only  when  the  needle  is  parallel  to  the  bar. 

At  intermediate  points  the  needle  will  assume  all  possible  in- 
clinations to  the  axis  of  the  bar,  each  position  being  determined 
by  the  resultant  of  the  four  forces  which  act  on  the  needle.  In 
Fig.  215  are  indicated  some  of  the  positions  which  the  needle  takes 

FIG.  215. 


in  being  carried  round  the  magnet.    While  it  goes  once  round  the 
magnet,  it  makes  two  revolutions  on  its  own  axis. 

It  is  to  be  observed  that  in  all  positions  the  needle  tends,  as  a 


MAGNETIC    CURVES. 


231 


whole,  to  move  toward  the  bar,  since  the  attractions  always  exceed 
the  repulsions. 

368.  Magnetic  Curves. — All  the  foregoing  cases  are  shown 
at  once  by  iron  filings  strewn  on  paper  or  parchment,  which  is 
stretched  on  a  frame  and  placed  near  a  magnet.  Let  the  paper  be 
slightly  jarred,  while  the  magnet  lies  parallel  to  it,  either  above  or 
below,  and  all  the  inclinations  of  the  needle  will  be  represented 
by  the  particles  of  iron  arranged  in  curves  from  pole  to  pole  (Fig. 
216).  Near  the  poles  of  the  magnet  the  filings  stand  up  on  the 

FIG.  216. 


paper  at  various  inclinations.  These  are  the  extremities  of  still 
other  curves,  which  would  be  formed  in  all  possible  planes  passing 
through  the  axis  of  the  magnet,  provided  the  filings  could  float 
suspended  in  the  air,  while  the  magnet  is  placed  in  the  midst  of 
them.  These  are  called  magnetic  curves. 

When  the  magnet  is  below  the  paper,  the  particles  move  away 
from  the  area  over  the  poles,  as  in  Fig.  216  ;  but  when  it  is  above, 
they  gather  in  a  cluster  under  each  pole.  This  singular  difference 
arises  from  the  force  of  gravity  acting  on  the  filaments,  which  are 
raised  up  on  the  paper,  and  which  lean,  in  the  former  case,  from 
each  other,  and  in  the  latter,  toward  each  other. 


CHAPTER   II. 

RELATIONS  OF  THE  MAGNET  TO  THE  EARTH. 

369.  Declination  of  the  Needle. — When  the  needle  is  bal- 
anced horizontally,  and  free  to  revolve,  it  does  not  generally  point 
exactly  north  and  south ;  and  the  angle  by  which  it  deviates  from 
the  meridian  is  called  the  declination.  A  vertical  circle  coinci- 


232 


MAGNETISM. 


dent  with  the  direction  of  the  needle  at  any  place  is  called  the 
magnetic  meridian.  As  the  angle  between  the  magnetic  and  the 
geographical  meridians  is  generally  different  for  different  places, 
and  also  varies  at  different  times  in  the  same  place,  the  word  vari- 
ation expresses  these  changes  in  declination,  though  it  is  much 
used  as  synonymous  with  declination  itself. 

370.  Isogonic  Curves. — This  name  is  given  to  a  system  of 
lines  imagined  to  be  drawn  through  all  the  points  of  equal  decli- 
nation on  the  earth's  surface.  We  naturally  take  as  the  standard 
line  of  the  system  that  which  connects  the  points  of  no  declina- 
tion, or  the  isogonic  of  0°  (Fig.  217).  Commencing  at  the  north 

FIG.  217. 


pole  of  dip,  about  Lat.  70°,  Lon.  96°,  it  runs  in  a  general  direc- 
tion E.  of  S.,  through  Hudson's  Bay,  across  Lake  Erie,  and  the 
State  of  Pennsylvania,  and  enters  the  Atlantic  Ocean  on  the  coast 
of  North  Carolina,  Thence  it  passes  east  of  the  West  India 
Islands,  and  across  the  N.  E.  part  of  South  America,  pursuing  its 
course  to  the  south  polar  regions.  It  reappears  in  the  eastern  hem- 
isphere, crosses  Western  Australia,  and  bears  rapidly  westward 
across  the  Indian  Ocean,  and  then  pursues  a  northerly  course 
across  the  Caspian  Sea  to  the  Arctic  Ocean.  There  is  also  a  de- 
tached line  of  no  declination,  lying  in  eastern  Asia  and  the  Pacific 
Ocean,  returning  into  itself,  and  inclosing  an  oval  area  of  40°  N". 
and  S.  by  30°  E.  and.  W.  Between  the  two  main  lines  of  no  decli- 
nation in  the  Atlantic  hemisphere,  the  declination  is  westward, 
marked  by  continued  lines,  in  Fig.  217 ;  in  the  Pacific  hemisphere, 
outside  of  the  oval  line  just  described,  it  is  eastward,  marked  by 
dotted  lines.  Hence,  on  the  American  continent,  in  all  places 
east  of  the  isogonic  of  0°,  the  north  pole  of  the  needle  declines 
westward,  and  in  all  places  west  of  it,  the  north  pole  declines  east- 


ANNUAL    AND    DIURNAL    VARIATIONS.  333 

ward ;  on  the  other  continent  this  is  reversed,  as  shown  by  the 
figure. 

Among  other  irregularities  in  the  isogonic  system,  there  are 
two  instances  in  which  a  curve  makes  a  wide  sweep,  and  then  in- 
tersects its  own  path,  while  those  within  the  loop  thus  formed  re- 
turn into  themselves.  One  of  these  is  the  isogonic  of  8°  40'  E., 
which  intersects  in  the  Pacific  Ocean  west  of  Central  America ; 
the  other  is  that  of  22°  13'  W.,  intersecting  in  Africa. 

In  the  northeastern  part  of  the  United  States  the  declination 
has  long  been  a  few  degrees  to  the  west,  with  very  slow  and  some- 
what irregular  variations. 

371.  Secular  and  Annual  Variation. — The  declination  of 
the  needle  at  a  given  place  is  not  constant,  but  is  subject  to  a 
slow  change,  which  carries  it  to  a  certain  limit  on  one  side  of  the 
meridian,  when  it  becomes  stationary  for  a  time,  and  then  returns, 
and  proceeds  to  a  certain  limit  on  the  other  side  of  it,  occupying 
two  or  three  centuries  in  each  vibration.    At  London,  in  1580, 
the  declination  was  11|°  E. ;  in  1657,  it  was  0° ;  after  which  time 
the  needle  continued  its  western  movement  till  1814,  when  the 
declination  was  24\°  W. ;  since  then  the  needle  has  been  moving 
slowly  eastward.    The  entire  secular  vibration  will  probably  last 
more  than  three  centuries.    The  average  variation  from  1580  to 
1814  was  9'  10"  annually.    But  like  other  vibrations,  the  motion 
is  slowest  toward  the  extremes. 

There  has  also  been  detected  a  small  annual  variation,  in  which 
the  needle  turns  its  north  pole  a  few  minutes  to  the  east  of  its 
mean  position  between  April  and  July,  and  to  the  west  the  rest 
of  the  year.  This  annual  oscillation  does  not  exceed  15  or  18 
minutes. 

372.  Diurnal  Variation. — The  needle  is  also  subject  to  a 
small  daily  oscillation.    In  the  morning  the  north  end  of  the 
needle  has  a  variation  to  the  east  of  its  mean  position  greater  than 
at  any  other  part  of  the  day.     During  winter  this  extreme  point  is 
attained  at  about  8  o'clock,  but  as  early  as  7  o'clock  in  the  sum- 
mer.   After  reaching  this  limit  it  gradually  moves  to  the  west, 
and  attains  its  extreme  position  at  about  3  o'clock  in  winter,  and 
1  o'clock  in  summer.    From  this  time  the  needle  again  returns 
eastward,  reaching  its  first  position  about  10  p.  M.,  and  is  almost 
stationary  during  the  night.     The  whole  amount  of  the  diurnal 
variation  rarely  exceeds  12  minutes,  and  is  commonly  much  less 
than  that.     These  diurnal  changes  of  declination  are  connected 
with  changes  of  temperature,  being  much  greater  in  summer  than 
in  winter.     Thus,  in  England  the  mean  diurnal  variation  from 
May  to  October  is  10  or  12  minutes,  and  from  November  to  April, 
only  5  or  6  minutes. 


234 


MAGNETISM. 


FIG 


373.  Dip  of  the  Needle. — A  needle  first  balanced  on  a  hor- 
izontal axis,  and  then  magnetized  and  placed  in  the  magnetic  meridi- 
an, assumes  a  fixed  relation  to 

the  horizon,  one  pole  or  the 
other  being  usually  depressed 
below  it.  The  angle  of  de- 
pression is  called  the  dip  of 
the  needle.  Fig.  218  repre- 
sents the  dipping  needle,  with 
its  adjusting  screws  and 
spirit-level;  and  the  depres- 
sion may  be  read  on  the  grad- 
uated scale.  After  the  hori- 
zontal circle  m  is  leveled  by 
the  foot-screws,  the  frame  A 
is  turned  horizontally  till  the 
vertical  circle  M  is  in  the 
magnetic  meridian.  For 
north  latitudes,  the  north 
end  of  the  needle  is  depressed, 
as  a  in  the  figure. 

374.  Is o clinic  Curves. 

— A  line  passing  through  all 

points  where  the  dip  of  the  needle  is  nothing,  i.  e.  where  the  dip- 
ping needle  is  horizontal,  is  called  the  magnetic  equator  of  the 
earth.  It  can  be  traced  in  Fig.  219  as  an  irregular  curve  around 

FIG.  219. 


the  earth  in  the  region  of  the  equator,  nowhere  departing  from  it 
more  than  about  15°.  At  every  place  north  of  the  magnetic  equa- 
tor the  north  pole  of  the  needle  descends,  and  south  of  it  the  south 
pole  descends;  and,  in  general,  the  greater  the  distance,  the 


MAGNETIC  INTENSITY   OF  THE  EARTH          335 

greater  is  the  dip.  Imagine  now  a  system  of  lines,  each  passing 
through  all  the  points  of  equal  dip ;  these  will  be  nearly  parallel 
to  the  magnetic  equator,  which  may  be  regarded  as  the  standard 
among  them.  These  magnetic  parallels  are  called  the  isoclinic 
curves  ;  they  somewhat  resemble  parallels  of  latitude,  but  are  in- 
clined to  them,  conforming  to  the  oblique  position  of  the  magnetic 
equator.  In  the  figure,  the  broken  lines  show  the  dip  of  the  south 
pole  of  the  needle ;  the  others,  that  of  the  north  pole.  The  points 
of  greatest  dip,  or  dip  of  90°,  are  called  the  poles  of  dip.  There  is 
one  in  the  northern  hemisphere,  and  one  in  the  southern.  The 
north  pole  of  dip  was  found,  by  Capt.  James  C.  Ross,  in  1831,  to 
be  at  or  very  near  the  point,  70°  14'  K ;  96°  40'  "W.,  marked  x  in 
the  figure.  The  south  pole  is  not  yet  so  well  determined. 

At  the  poles  of  dip  the  horizontal  needle  loses  all  its  directive 
power,  because  the  earth's  magnetism  tends  to  place  it  in  a  verti- 
cal line,  and,  therefore,  no  component  of  the  force  can  operate  in 
a  horizontal  plane.  The  isogonic  lines  in  general  converge  to  the 
two  dip-poles ;  but,  for  the  reason  just  given,  they  cannot  be  traced 
quite  to  them. 

The  dip  of  the  needle,  like  the  declination,  undergoes  a  varia- 
tion, though  by  no  means  to  so  great  an  extent.  In  the  course 
of  250  years,  it  has  diminished  about  five  degrees  in  London.  In 
1820  it  was  about  70°,  and  diminishes  from  two  to  three  minutes 
annually. 

Since  the  dip  at  a  given  place  is  changing,  it  cannot  be  sup- 
posed that  the  poles  are  fixed  points ;  they,  and  with  them  the  en- 
tire system  of  isoclinic  curves,  must  be  slowly  shifting  their  lo- 
cality. 

375.  Magnetic  Intensity  of  the  Earth. — The  force  ex- 
erted by  the  magnetism  of  the  earth  varies  in  different  places, 
being  generally  least  in  the  region  of  the  equator,  and  greatest  in 
the  polar  regions.     The  ratio  of  intensity  in  different  places  is 
measured  by  the  number  of  vibrations  which  the  needle  makes  in 
a  given  time.    In  the  discussion  of  the  pendulum,  it  was  proved 
(Art.  170)  that  gravity  varies  as  the  square  of  the  number  of  vi- 
brations.   For  the  same  reason,  the  magnetic  force  at  any  place 
varies  as  the  square  of  the  number  of  vibrations  of  the  needle  at 
that  place. 

376.  Isodynamic  Curves. — After  ascertaining,  by  actual 
observation,  the  intensity  of  the  magnetic  force  in  different  parts 
of  the  earth,  lines  are  supposed  to  be  drawn  through  all  those 
points  in  which  the  force  is  the  same;  these  lines  are  called  isody- 
namic  curves,  represented  in  Fig.  220.    These  also  slightly  re- 
semble  parallels  of  latitude,  but  are  more  irregular  than  the 


236  MAGNETISM. 

isoclinic  lines.  There  is  no  one  standard  equator  of  minimum  in- 
tensity, but  there  are  two  very  irregular  lines  surrounding  the 
earth  in  the  equatorial  region,  in  some  places  almost  meeting  each 


FIG.  220. 


other,  and  in  others  spreading  apart  more  than  two  thousand 
miles,  on  which  the  magnetic  intensity  is  the  same.  These  two 
are  taken  as  the  standard  of  comparison,  because  they  are  the 
lowest  which  extend  entirely  round  'the  globe.  The  intensity  on 
them  is  therefore  called  unity,  marked  1  in  the  figure.  In  the 
wide  parts  of  the  belt  which  they  include — lying  one  in  the  south- 
ern Atlantic,  and  the  other  in  the  northern  Pacific  oceans — there 
are  lines  of  lower  intensity  which  return  into  themselves,  without 
encompassing  the  earth.  In  approaching  the  polar  regions,  both 
north  and  south,  the  curves,  retaining  somewhat  the  form  of 
the  unit  lines,  are  indented  like  an  hour-glass,  as  those  marked 
1.7  in  the  figure,  and  at  length  the  indentations  meet,  forming  an 
irregular  figure  8 ;  and  at  still  higher  latitudes,  are  separated  into 
two  systems,  closing  up  around  two  poles  of  maximum  intensity. 
Thus  there  are  on  the  earth  four  poles  of  maximum  intensity,  two 
in  the  northern  hemisphere  and  two  in  the  southern.  The  Amer- 
ican north  pole  of  intensity  is  situated  on  the  north  shore  of  Lake 
Superior.  The  one  on  the  eastern  continent  is  in  northern- Si- 
beria. The  ratio  of  the  least  to  the  greatest  intensity  on  the  earth 
is  about  as  0.7  to  1.9 ;  that  is,  as  1  to  2|.  In  the  figure,  intensities 
less  than  1  are  marked  by  dotted  lines. 

377.  Magnetic  Charts.— These  are  maps  of  a  country,  or  of 
the  world,  on  which  are  laid  down  the  systems  of  curves  which 
have  been  described.  But  for  the  use  of  the  navigator,  only  the 
isogonic  lines,  or  lines  of  equal  declination,  are  essential.  There 
are  large  portions  of  the  globe  which  have  as  yet  been  too  imper- 


MAGNETIC    OBSERVATORIES.  237 

fectly  examined  for  the  several  systems  of  curves  to  be  accurately 
mapped.  It  must  be  remembered,  too,  that  the  earth  is  slowly 
but  constantly  undergoing  magnetic  changes,  by  which,  at  any 
given  place,  the  declination,  dip,  and  intensity  are  all  essentially 
altered  after  the  lapse  of  years.  A  chart,  therefore,  which  would 
be  accurate  for  the  middle  of  the  nineteenth  century,  will  be,  to 
some  extent,  incorrect  at  its  close. 

378.  Magnetic  Observatories. — In  accordance  with  a  sug- 
gestion of  Humboldt,  in  1836,  systematic  observations  have  been 
since  made  upon  terrestrial  magnetism,  in  various  parts  of  the 
world,  in  order  to  deduce  from  them  the  laws  of  its  changes. 
Buildings  have  been  erected  without  any  iron  in  their  construc- 
tion, to  serve  as  magnetic  observatories ;  and  the  most  delicate 
magnetometers  have  been  devised  and  used  for  detecting  minute 
oscillations  both  in  the  horizontal  and  vertical  planes.     By  these 
means  has  been  discovered  a  class  of  phenomena  called  magnetic 
storms,  in  which  the  needle  suffers  numerous  and  rapid  disturb- 
ances, sometimes  to  the  extent  of  several  degrees ;  and  it  is  a  re- 
markable and  interesting  fact  that  these  disturbances  occur  at  the 
same  absolute  time  in  every  part  of  the  earth. 

379.  Aurora  Borealis. — This  phenomenon  is  usually  ac- 
companied by  a  disturbance  of  the  needle,  thus  affording  visible 
indications  of  a  magnetic  storm ;  but  the  contrary  is  by  no  means 
generally  true,  that  a  magnetic  storm  is  accompanied  by  auroral 
light.     The  connection  of  the  aurora  borealis  with  magnetism  is 
manifested  not  only  by  the  disturbance  of  the  needle,  but  also  by 
the  fact  that  the  streamers  are  parallel  to  the  dipping  needle,  as  is 
proved  by  their  apparent  convergence  to  that  point  of  the  sky  to 
which  the  dipping-needle  is  directed.    This  convergence  is  the 
effect  of  perspective,  the  lines  being  in  fact  straight  and  parallel. 

380.  Source  of  the  Earth's  Magnetism. — If  a  needle  is 
carried  round  the  earth  from  north  to  south,  it  takes  approxi- 
mately all  the  positions  in  relation  to  the  earth's  axis  which  it  as- 
sumes in  relation  to  a  magnetic  bar,  when  carried  round  it  from 
end  to  end  (Art.  367).    At  the  equator  it  is  nearly  parallel  to  the 
axis,  and  it  inclines  at  larger  and  larger  angles  as  the  distance 
from  the  equator  increases ;  and  in  the  region  of  the  poles,  it  is 
nearly  in  the  direction  of  the  axis.     The  earth  itself,  therefore, 
may  be  considered  a  magnet,  since  it  affects  a  needle  as  a  magnet 
does,  and  also  induces  the  magnetic  state  on  iron.    But  it  is  nec- 
essary, on  account  of  the  attraction  of  opposite  poles,  to  consider 
the  northern  part  of  the  earth  as  being  like  the  south  pole  of  a 
needle,  and  the  southern  part  like  the  north  pole.    To  avoid  this, 


238  MAGNETISM. 

the  words  boreal  and  austral  are  applied  to  the  two  magnetic 
states,  and  the  boreal  magnetism  is  the  name  given  to  that  devel- 
opment found  in  the  northern  hemisphere,  and  the  austral  mag- 
netism to  that  in  the  southern.  Hence,  it  becomes  necessary,  in 
using  these  names  for  a  magnet,  to  reverse  their  order,  and  to 
speak  of  its  north  pole  as  exhibiting  the  austral,  and  its  south  pole 
the  boreal  magnetism. 

Modern  discoveries  in  electro-magnetism  and  thermo-electri- 
city furnish  a  clew  to  the  hypothesis  which  generally  prevails  at 
this  day.  Attention  has  been  drawn  to  the  remarkable  agreement 
between  the  isothermal  and  the  isomagnetic  lines  of  the  globe. 
The  former  descend  in  crossing  the  Atlantic  Ocean  toward  Amer- 
ica, and  there  are  two  poles  of  maximum  cold  in  the  northern 
hemisphere.  The  isoclinic  and  the  isodynamic  curves  also  de- 
scend to  lower  latitudes  in  crossing  the  Atlantic  westward ;  so 
that,  at  a  given  latitude,  the  degree  of  cold,  the  magnetic  dip,  and 
the  magnetic  intensify,  is  each  considerably  greater  on  the  Amer- 
ican than  on  the  European  coast.  This  is  only  an  instance  of  the 
general  correspondence  between  these  different  systems  of  curves. 
It  has  likewise  been  noticed  (Art.  372)  that  the  needle  has  a  move- 
ment diurnally,  varying  westward  during  the  middle  of  the  day, 
and  eastward  at  evening,  and  that  this  oscillation  is  generally 
much  greater  in  the  hot  season  than  the  cold.  It  is  obvious, 
therefore,  that  the  development  of  magnetism  in  the  earth  is  inti- 
mately connected  with  the  temperature  of  its  surface.  Hence  it  is 
supposed  that  the  heat  received  from  the  sun  excites  electric  cur- 
rents in  the  materials  of  the  earth's  surface,  and  these  give  rise  to 
the  magnetic  phenomena. 

381.  Formation  of  Permanent  Magnets. — Needles  and 
small  bars  may  be  more  or  less  magnetized  by  the  following  meth- 
ods, the  reasons  for  which  will  be  readily  understood : 

1.  A  feeble  magnetism  may  be  developed  in  a  steel  bar,  by 
causing  it  to  ring  while  held  vertically.     The  earth's  influence 
upon  it,  however,  is  stronger  if  it  is  held,  not  precisely  vertical, 
but  leaning  in  a  direction  parallel  to  the  dipping  needle.    The 
inductive  influence  of  the  earth  explains  the  fact  often  noticed, 
that  rods  of  iron  or  steel  that  have  stood  for  many  years  in  a  posi- 
tion nearly  vertical,  as,  for  instance,  lightning-rods,  iron  pillars, 
stoves,  &c.,  are  found  somewhat  magnetic,  with  the  north  pole 
downward. 

2.  A  needle  may  be  magnetized  by  simply  suffering  it  to  re- 
main in  contact  with  the  pole  of  a  strong  magnet,  or  better,  be- 
tween the  opposite  poles  of  two  magnets. 

3.  Place  the  needle  across  the  opposite  poles  of  two  parallel 


THE    COMPASS. 


239 


magnets,  while  a  bar  of  soft  iron  connects  the  other  two  poles. 
Thus,  removing  one  of  the  keepers,  A,  B,  from  the  ends  of  the 
magnets  (Fig.  221),  put  the  needle  in  its  place,  being  careful  that 
the  end  of  the  needle 
marked  for  north  is  ad- 
jacent to  the  south  pole 
of  the  magnet. 

4.  In  order  to  take 
advantage  of  the  earth's 
inductive  influence,  along  with  that  of  steel  magnets,  place  the 
needle  parallel  to  the  dipping  needle,  and  draw  the  south  pole  of 
one  magnet  over  the  lower  half,  and  the  north  pole  of  another 
over  the  upper  half,  with  repeated  and  simultaneous  move- 
ments. 

None  of  these  methods,  however,  are  of  great  practical  value  at 
the  present  day,  since  the  galvanic  circuit  affords  a  far  readier  and 
more  efficient  means  of  magnetizing  bars. 

The  horse-shoe  magnet,  sometimes  called  the  FIG.  222. 
U-magnet  (Fig.  222),  is  for  many  purposes  a  very 
convenient  form,  and  originated  in  the  practice  of 
arming  the  lodestone;  that  is,  furnishing  it  with  two 
pieces  of  soft  iron,  which  are  confined  by  brass  straps 
to  the  poles  of  the  stone,  and  project  below  it,  so 
that  a  bar  and  weight  may  be  attached.  "When  a 
magnet  has  this  form,  both  poles  may  be  applied  to  a 
body  at  once.  The  Z7-magnet,  A  N  S,  being  sus- 
pended, and  the  keeper,  B,  made  of  soft  iron,  being 
attached  to  the  poles,  weights  may  be  hung  upon 
the  hook  C,  to  show  the  strength. 

382.  The  Declination  Compass.— This  in- 
strument consists  of  a  magnetic  needle  suspended  in 
the  centre  of  a  cylindrical  brass  box  covered  with 
glass ;  on  the  bottom  of  the  box  within  is  fastened  a 
circular  card,  divided  into  degrees  and  minutes,  from 
0°  to  90°  on  the  several  quadrants.  On  the  top  of  ^ 

the  box  are  two  uprights,  either  for  holding  sight- 
lines  or  for  supporting  a  small  telescope,  by  which  directions  are 
fixed.    The  quadrants  on  the  card  in  the  box  are  graduated  from 
that  diameter  which  is  vertically  beneath  the  line  of  sight. 

When  the  axis  of  vision  is  directed  along  a  given  line,  the 
needle  shows  how  many  degrees  that  line  is  inclined  to  the  mag- 
netic meridian.  In  o^der  that  the  angle  between  the  line  and  the 
geographical  meridian  may  be  found,  the  declination  of  the  needle 
for  the  place  must  be  known. 


240  MAGNETISM. 

383.  The  Mariner's  Compass.— In  the  mariner's  compass 
(Fig.  223)  the  card  is  made  as  light  as  possible,  and  attached  to  the 
needle,  so  that  the  north  and  south 

points  marked  on  th'e  card  always 
coincide  with  the  magnetic  merid- 
ian. The  index,  by  which  the  di- 
rection of  the  ship  is  read,  consists 
of  a  pair  of  vertical  lines,  diametri- 
cally opposite  to  each  other,  on  the 
interior  of  the  box.  These  lines, 
one  of  which  is  seen  at  a,  are  in  the 
plane  of  the  ship's  keel.  Hence, 
the  degree  of  the  card  which  is  against  either  of  the  lines  shows 
at  once  both  the  angle  with  the  magnetic  meridian  and  the  quad- 
rant in  which  that  angle  lies. 

In  order  that  the  top  of  the  box  may  always  be  in  a  horizontal 
position,  and  the  needle  as  free  as  possible  from  agitation  by  the 
rolling  of  the  ship,  the  box,  B,  is  suspended  in  gimbals.  The 
pivots,  A,  A,  on  opposite  sides  of  the  box,  are  centred  in  the  brass 
ring,  (7,  Z),  while  this  ring  rests  on  an  axis,  which  has  its  bearings 
in  the  supports,  E,  E.  These  two  axes  are  at  right  angles  to  each 
other,  and  intersect  at  the  point  where  the  needle  rests  on  its 
pivot.  Therefore,  whatever  position  the  supports,  E,  E,  may  have, 
the  box,  having  its  principal  weight  in  the  lower  part,  maintains 
its  upright  position,  and  the  centre  of  the  needle  is  not  moved  by 
the  revolutions  on  the  two  axes. 

On  account  of  the  dip,  which  increases  with  the  distance  from 
the  equator,  and  is  reversed  by  going  from  one  hemisphere  to  the 
other,  the  needle  needs  to  be  loaded  by  a  small  adjustable  weight, 
if  it  is  to  be  used  in  extensive  voyages  to  the  north  or  south.  In 
north  latitudes  the  south  end  must  be  heaviest;  in.  south  lati- 
tudes, the  north  end. 

384.  The  Needle  Rendered  Astatic. — Though  magnetic 
intensity  increases  at  greater  distances  from  the  equator,  yet  the 
directive  power  of  the  compass  grows  more  feeble  in  approaching 
the  poles  of  dip,  because  the  horizontal  component  constantly  di- 
minishes, and  at  the  poles  becomes  zero  (Art.  374).    A  needle  in 
such  a  situation,  in  which  the  earth's  magnetism  has  no  influence 
to  give  it  direction,  is  called  astatic.   The  compass  needle  is  astatic 
at  the  north  and  south  poles  of  dip.    And  the  dipping  needle 
may  be  rendered  astatic  at  any  place  by  setting  its  plane  of  rota- 
tion perpendicular  to  its  line  of  dip  at  that*place  ;  for  then  there 
will  remain  no  component  of  the  magnetic  force-  in  the  only  plane 
in  which  the  needle  is  at  liberty  to  move. 


THEORY    OF    MAGNETISM.  241 

The  needle  may  also  be  made  astatic  at  any  place  by  holding  a 
magnet  at  such  a  distance,  and  in  such  a 
position,  as   to  neutralize  the  earth's  in-  ^IG.  224 

fluence.  Or,  if  a  wire,  suspended  verti- 
cally by  a  thread,  pass  through  the  centres 
of  two  needles,  whose  poles  point  in  oppo- 
site directions,  each  needle  will  be  astatic. 
The  needles  in  Fig.  224,  with  like  poles  in 
opposite  directions,  are  slipped  tightly  upon 
the  wire  #  c,  which  is  suspended  by  the 
thread  a  b,  free  from  torsion.  This  method 
of  liberating  a  magnetic  needle  from  the 
earth's  influence  is  of  great  use  in  electro-magnetism. 

385.  Theory  of  Magnetism. — The  nature  of  the  agency 
called  magnetism  is  unknown.  Much  of  the  language  employed 
by  writers  on  the  subject  implies  that  there  exist  in  iron,  steel, 
&c.,  two  imponderable  fluids,  called  the  austral  and  boreal  magnet- 
isms ;  that  these  fluids  attract  each  other,  and  are  ordinarily 
mingled  and  neutralized,  so  that  no  magnetic  phenomena  appear ; 
and  that  in  every  magnet  the  two  fluids  have  been  separated  by 
the  inductive  influence  of  the  earth  or  of  another  magnet,  one 
fluid  manifesting  itself  at  one  pole,  and  the  other  at  the  other 
pole.  As  science  advances,  however,  these  views  seem  more  and 
more  crude  and  unsatisfactory.  Magnetism  is  now  regarded  by 
many  as  one  of  those  modes  of  molecular  motion  which  are  so  diffi- 
cult of  investigation.  If  it  is  a  mode  of  motion,  then  it  may  man- 
ifest itself  as  a  force,  as  we  know  it  does.  It  will  be  seen  in  the 
discussion  of  Electro-magnetism  that  there  is  a  most  intimate 
connection  between  magnetism  and  electricity,  so  much  so  that 
the  former  is  generally  considered  as  only  a  particular  form  in 
which  the  latter  is  developed. 

Magnetism  differs  from  the  other  molecular  agencies — elec- 
tricity, light,  and  heat — in  producing  no  direct  effect  on  any  of 
our  senses.    We  witness  its  direct  effects  only  in  the  motion  which 
it  gives  to  certain  kinds  of  matter,  such  as  iron  and  steel. 
16 


PART   VI. 

FKICTIONAL    OE   STATICAL   ELECTRICITY, 


CHAPTER  I. 

ELEMENTARY    PHENOMENA. 

386.  Definitions. — The  name  Electricity,  from  the  Greek 
word  for  amber,  is  given  to  a  peculiar  agency,  which  is  the  cause 
of  a  variety  of  phenomena,  such  as  attracting  and  repelling  light 
bodies,  producing  light,  heat,  sound,  and  chemical  decomposition, 
and,  when  concentrated  in  its  action,  violently  rending  or  explod- 
ing bodies.    Lightning  and  thunder  are  an  example  of  its  intense 
action. 

Fractional  electricity  is  so  called  because  generally  excited  by 
friction,  and  to  distinguish  this  form  of  development  from  the  gal- 
vanic electricity  which  is  excited  by  chemical  means.  The  former 
is  often  called  statical,  and  the  latter  dynamical  electricity. 

Bodies  are  said  to  be  electrically  excited  when  they  show  signs 
of  electricity  by  some  action  performed  upon  them,  as  friction,  for 
example.  They  are  said  to  be  electrified  when  they  receive  elec- 
tricity by  communication. 

Conductors  are  bodies  which  transmit  electricity  freely ;  non- 
conductors are  those  which  do  not  transmit  it  at  all,  or  only  very 
imperfectly.  A  body  is  said  to  be  insulated  when  in  contact  only 
with  non-conductors,  so  that  electricity  is  retained  in  it. 

387.  Electroscopes. — The  feeblest  indication  of  electricity 
is  usually  attraction  or  repulsion ;  and  instruments  prepared  for 
showing  these  effects  are  called  electroscopes.     The  word  electrome- 
ter, though  sometimes  used  in  the  same  sense,  is  more  properly 
defined  to  be  an  instrument  for  measuring  the  quantity  of  elec- 
tricity. 

The  pendulum  electroscope  (Fig.  225)  consists  of  a  glass  stand- 
ard, supported  by  a  base,  and  bent  into  a  hook  at  the  top,  from 
which  is  suspended  a  pith  ball  by  a  fine  silk  thread. 


INDICATIONS    OP    ELECTRICITY. 


243 


The  gold-leaf  electroscope  consists  of  two  narrow  strips  of  gold- 
leaf,  n,  n  (Fig.  226),  suspended  within  a  glass  receiver,  B,  from  a 


FIG.  225. 


FIG.  226. 


metallic  rod  which  passes  through  the  top  and  terminates  in  a 
ball,  C.  A  metallic  base  is  cemented  to  the  receiver,  and  strips  of 
tin-foil,  a,  are  attached  to  the  inside,  reaching  to  the  base.  When 
an  electrified  body  is  brought  near  the  knob  C9  the  gold  leaves 
separate,  or,  if  separated,  collapse,  or  separate  more,  according  to 
circumstances. 

Certain  modifications  are  convenient  for  some  purposes.  One 
is,  a  metallic  wire  with  a  ball,  on  the  top,  having  a  thread  and 
pith  ball  hanging  by  the  side  of  it;  and  another,  two  threads  with 
pith  balls  suspended  together  below  a  conductor,  as  in  Fig.  231. 

388.  Common  Indications  of  Electricity. — Though  the 
frictional  or  statical  electricity  may  be  developed  in  several  ways, 
pressure,  evaporation,  &c.,  the  method  generally  employed  is  fric- 
tion. By  this  means  it  can  be  excited  in  a  greater  or  less  degree 
in  all  substances,  and  from  some  it  may  be  easily  and  abundantly 
obtained. 

If  amber,  sealing-wax,  or  any  other  resinous  substance,  be 
rubbed  with  dry  woolen  cloth,  fur,  or  silk,  and  then  brought  near 
the  face,  the  excited  electricity  disturbs  the  downy  hairs  upon  the 
skin,  and  thus  causes  a  sensation  like  that  produced  by  a  cobweb. 
When  the  tube  is  strongly  excited,  it  gives  off  a  spark  to  the  finger 
held  toward  it,  accompanied  by  a  sharp  snapping  noise.  A  sheet 
of  writing-paper,  first  dried  by  the  fire,  and  then  laid  on  a  table 
and  rubbed  with  india-rubber,  becomes  so  much  excited  as  to  ad- 
here to  the  wall  of  the  room  or  any  other  surface  to  which  it  is 
applied.  As  the  paper  is  pulled  up  slowly  from  the  table  by  one 
edge,  a  number  of  small  sparks  may  be  seen  and  heard  on  the 


244  STATICAL    ELECTRICITY. 

under  side  of  the  paper.     In  dry  weather,  the  brushing  of  a  gar- 
ment causes  the  floating  dust  to  fly  back  and  cling  to  it. 

If  an  iron  or  brass  rod  be  held  in  the  hand  and  rubbed  with 
silk,  the  rod  shows  no  sign  of  electricity.  It  will  be  seen  here- 
after that  the  electricity  excited  in  the  rod  is  conveyed  away  by 
the  conducting  quality  of  the  metal  and  the  human  body. 

389.  The   Two   Electrical   States. — When    friction  has 
taken  place  between  two  bodies,  they  are  found  in  electrical  condi- 
tions, which  in  some  remarkable  particulars  are  unlike  each  other. 
These  two  electrical  states  are  usually  called  the  positive  and  the 
negative,  terms  which  were  employed  by  Franklin  in  his  theory  of 
one  electric  fluid,  to  indicate  that  the  excited  body  has  either  more 
or  less  electricity  than  belongs  to  it  in  its  common  unexcited  con- 
dition.   Du  Fay,  in  his  theory  of  two  kinds  of  electricity,  uses  the 
words  vitreous  and  resinous  to  distinguish  them,  vitreous  corre- 
sponding to  the  positive,  and  resinous  to  the  negative.    But  it  is 
very  common  to  use  Du  Fay's  theory,  and  to  apply  Franklin's 
terms,  positive  and  negative,  to  the  two  kinds  of  electricity. 

If  in  any  case  only  one  electricity  is  discovered  when  friction 
causes  development,  it  is  to  be  understood  that  the  other  is  dif- 
fused through  some  large  conductor,  so  as  to  be  imperceptible. 
The  earth  is  the  great  reservoir,  in  which  any  amount  of  elec- 
tricity may  be  diffused  and  lost  sight  of. 

390.  Nature  of  Electricity. — The  real  nature  of  electricity 
is  unknown.    Though  it  is  in  most  treatises  spoken  of  as  a  fluid, 
of  exceeding  rarity,  and  more  rapid  in  its  movements  than  light, 
yet  the  prevailing  belief  at  the  present  day  is,  that  it  is  a  peculiar 
mode  of  vibratory  motion,  either  in  the  luminiferous  ether  which 
is  imagined  to  fill  all  space,  or  else  in  the  ordinary  matter  consti- 
tuting the  bodies  and  media  about  us,  or  in  both  of  these.    Elec- 
tricity is  brought -to  view  by  friction,  by  heat,  and  by  other 
agencies  which  are  calculated  to  cause  movements  in  matter,  rather 
than  to  bring  new  kinds  of  matter  to  light.     It  is  undoubtedly  one 
of  the  forms  of  force,  into  which  other  forces  may  be  transformed. 
But  until  a  more  definite  wave-theory  or  force-theory  can  be  con- 
structed than  exists  at  present,  it  is  comparatively  easy  to  give  to 
the  learner  an  intelligible  description  of  electrical  phenomena  by 
using  the  language  of  the  two-fluid  theory  of  Du  Fay.    In  trying 
to  give  a  statement  of  observed  facts  without  the  use  of  these  hy- 
pothetical terms,  it  is  necessary  to  employ  in  their  stead  tedious 
circumlocutions,  which  only  confuse  the  mind  of  the  learner. 

391.  Du  Fay's  Theory. — According  to  this  theory,  the  two 
fluids  are  imagined  to  inhere  in  all  kinds  of  matter,  combined  with 


THE    TWO    ELECTRICAL    STATES.  245 

each  other  and  neutralized.  In  this  condition,  they  afford  no  evi- 
dence of  their  existence.  But  they  can  in  several  ways  be  sepa- 
rated from  each  other;  and  when  thus  separated,  they  give  rise  to 
electrical  phenomena. 

392.  The   Two  States  Developed  Simultaneously. — 

If  bodies  are  rubbed  together,  the  two  electricities  are  separated, 
and  one  body  is  electrified  positively,  the  other  negatively.  For 
example,  glass  rubbed  with  silk  is  itself  positive,  and  the  silk  is 
negative.  But  the  same  substance  does  not  always  show  the  same 
kind  of  electricity,  since  that  depends  frequently  on  the  substance 
against  which  it  is  rubbed.  Dry  woolen  cloth  rubbed  on  smooth 
glass  is  negative,  but  on  sulphur  it  is  positive.  The  following 
table  contains  a  few  substances,  arranged  with  reference  to  this. 
Any  one  of  them,  rubbed  with  one  that  follows  it,  is  positively 
electrified  itself,  and  the  other  negatively : 

1.  Fur  of  a  cat.  7.  Silk. 

2.  Smooth  glass.  8.  Gum  lac. 

3.  Flannel.  9.  Resin. 

4.  Feathers.  10.  Sulphur. 

5.  Wood.  11.  India-rubber. 

6.  Paper.  12.  Gutta-percha. 

According  to  the  above  table,  silk  rubbed  on  smooth  glass  is 
negatively  excited ;  but  rubbed  on  sulphur,  it  is  excited  positively. 
It  is  sometimes  found,  however,  that  the  previous  electrical  condi- 
tion of  one  of  the  bodies  will  invert  the  order  stated  in  the  table. 
For  example,  if  silk,  having  been  rubbed  on  smooth  glass,  and 
therefore  being  negative,  should  then  be  rubbed  on  resin,  it  would 
probably  retain  its  negative  state,  and  the  resin  become  positively 
electrified,  contrary  to  the  order  of  the  table. 

The  mechanical  condition  of  the  surface  sometimes  changes 
the  order  of  the  two  electricities.  Thus,  if  glass  is  ground,  so  as 
to  lose  its  polish,  it  is  likely  to  be  negative  when  rubbed  with 
silk;  but  the  excitation  of  rough  glass  is  very  feeble. 

393.  Mutual  Action. — Bodies  electrified  in  different  ways 
attract,  and  in  the  same  ivay  repel  each  other.    Thus,  if  an  insu- 
lated pith  ball,  or  a  lock  of  cotton,  be  electrified  by  touching  it 
with  an  excited  glass  tube,  it  will  immediately  recede  from  the 
tube,  and  from  all  other  bodies  which  are  charged  with  the  posi- 
tive electricity,  while  it  will  be  attracted  by  excited  sealing-wax, 
and  by  all  other  bodies  which  are  negatively  electrified.    If  a  lock 
of  fine  long  hair  be  held  at  one  end,  and  brushed  with  a  dry  brush, 
the  separate  hairs  will  become  electrified,  and  will  repel  each  other. 
In  like  manner,  two  insulated  pith  balls,  or  any  other  light  bodies, 


246  STATICAL    ELECTRICITY. 

will  repel  each  other  when  they  are  electrified  the  same  way,  and 
attract  each  other  when  they  are  electrified  in  different  ways. 

Hence  it  is  easy  to  determine  whether  the  electricity  developed 
in  a  given  body  is  positive  or  negative ;  for,  having  charged  the 
electroscope  with  excited  glass,  then  all  those  bodies  which,  when 
excited,  attract  the  ball,  are  negative,  while  all  those  which  repel 
it  are  positive. 

394.  Conduction.— Electricity  passes  through  some  bodies 
with  the  greatest    facility;    through  others   with   difficulty,   or 
scarcely  at  all;  and  others  still  have  a  conducting  power  interme- 
diate between  the  two.    As  the  conducting  quality  exists  in  differ- 
ent substances  in  all  conceivable  degrees,  it  is  impossible  to  draw 
a  dividing  line  between  them,  so  as  to  arrange  all  conductors  on 
one  side,  and  all  non-conductors  on  the  other.    The  following 
brief  table  contains  some  of  the  more  important  of  the  two  classes ; 
the  first  column  in  the  order  of  conducting  power,  the  second  in 
the  order  of  insulating  power: 

Conductors.  Insulators. 

The  metals,  Lac,  amber,  the  resins, 

Charcoal,  Paraffine, 

Plumbago,  Sulphur, 

"Water,  damp  snow,  Wax, 

Living  vegetables,  Glass,  precious  stones, 

Living  animals,  Silk,  wool,  hair,  feathers, 

Smoke,  steam,  Paper, 

Moist  earth,  stones,  Air,  the  gases, 

Linen,  cotton.  Baked  wood. 

When  air  is  rarefied,  its  insulating  power  is  diminished,  and 
the  further  the  rarefaction  proceeds,  the  more  freely  does  elec- 
tricity pass.  Hence,  we  might  expect  that  it  would  pass  with  per- 
fect freedom  through  a  complete  vacuum.  It  is  found,  however, 
that  in  an  absolute  vacuum  electricity  cannot  be  transmitted 
at  all. 

395.  Modes  of  Insulating. — Solid  insulating  supports  are 
usually  made  of  glass ;  and,  in  order  to  improve  their  insulating 
power,  they  are  sometimes  covered  with  shell-lac  varnish.    Insu- 
lating threads  for  pith  balls,  or  cords  for  suspending  heavier 
bodies,  are  made  of  silk.    The  best  insulator  for  suspending  any 
very  small  weight  is  a  single  fiber  of  silk,  a  hair,  or  a  fine  thread 
of  gum  lac.    In  order  to  perform  electrical  experiments,  the  air 
must  be  dry,  or  no  care  whatever  relating  to  apparatus  can  insure 
success ;  and  therefore,  in  a  room  occupied  by  an  audience,  es- 
pecially if  the  weather  is  damp,  it  is  necessary  to  dry  the  air  arti- 


THE    PLATE    MACHINE.  247 

ficially  by  fires.    If  the  air  were  a  good  conductor,  it  is  probable 
that  no  facts  in  this  science  would  ever  have  been  discovered. 

396.  Communication  and  Influence.— The  sphere  of  com- 
munication is  the  space  within  which  a  spark  may  pass  from  an 
electrified  body,  in  any  direction.  It  is  sometimes  called  the 
striking  distance.  The  sphere  of  influence  is  the  space  within 
which  the  power  of  attraction  of  an  electrified  body  extends  every 
way,  beyond  the  sphere  of  communication.  A  glass  tube  strongly 
excited  will  give  motion  to  the  gold-leaf  electroscope  at  the  dis- 
tance of  several  feet,  although  a  spark  could  not  pass  from  the 
tube  to  the  cap  of  the  electroscope  at  a  greater  distance  than  a  few 
inches.  The  electricity  which  a  body  manifests  by  being  brought 
towards  an  excited  body,  without  receiving  a  spark  from  it,  is  said 
to  be  acquired  by  induction.  The  principle  of  induction  resembles 
that  noticed  in  magnetism,  and  will  be  discussed  in  connection 
with  the  Leyden  jar. 


CHAPTER    II. 

ELECTRICAL  MACHINES.— LAW  OF    FORCE.— MODE    OF    DISTRI- 
BUTION. 

397.  The  Plate  Machine. — In  order  that  glass  may  be  con- 
veniently subjected  to  friction  for  the  development  of  electricity, 
it  is  made  in  the  form  of  a  circular  plate,  and  mounted  on  an  axis, 
which  is  supported  by  a  wooden  frame,  and  revolved  by  a  crank, 
while  rubbers  press  against  its  surface.  Fig.  227  represents  one 
of  the  many  forms  which  have  been  adopted.  The  crank,  M,  gives 
rotary  motion  to  the  plate,  P,  which  is  pressed  by  the  rubbers, 
F,  F\  this  pressure  is  equalized  by  their  being  placed  at  top  and 
bottom,  and  on  both  sides  of  the  glass.  The  prime  conductor, 
C  C,  is  made  of  hollow  brass,  and  supported  by  glass  pillars.  The 
extremities  terminate  in  two  bows,  which  pass  around  the  edges 
of  the  plate,  and  present  to  it  a  few  sharp  points,  to  facilitate  the 
passage  of  electricity.  But  all  other  parts  are  carefully  rounded 
in  cylindrical  and  spherical  forms,  without  edges  or  points,  as 
these  tend  to  dissipate  the  electricity.  The  glass,  as  it  revolves 
from  the  rubbers  to  the  points  of  the  prime  conductor,  is  pro- 
tected by  silk  covers,  to  prevent  the  electricity  from  escaping  into 
the  air.  The  rubbers  are  made  of  soft  leather,  attached  to  a  piece 
of  wood  or  metal,  and  from  time  to  time  are  rubbed  over  with  an 
amalgam  of  zinc,  tin,  and  mercury,  or  with  the  bi-sulphuret  of 


248 


STATICAL    ELECTRICITY. 


tin,  which  is  one  of  the  best  exciters  on  glass.  The  diameter  of 
the  plate  varies  from  1£  to  3  feet;  but  in  some  of  the  largest  it  is 
6  feet,  and  two  plates  are  sometimes  mounted  on  one  axis. 

FIG.  227. 


To  giye  free  passage  of  the  negative  electricity  from  the  rub- 
bers to  the  earth,  a  chain,  D,  may  be  attached  to  the  wooden  sup- 
port, while  its  other  end  lies  on  the  floor. 

398.  The  Cylinder  Machine. — In  many  electrical  machines 
of  the  smaller  sizes,  a  hollow  cylinder  is  employed,  having  a  length 
considerably  exceeding  its  diameter.    In  the  cylinder  machine, 
the  rubber  is  applied  to  one  side,  and  the  prime  conductor  receives 
the  fluid  from  the  opposite.    The  rubber  is  usually  mounted  on  a 
glass  pillar,  so  that  it  can  be  insulated,  whenever  it  is  desired 

399.  The  Hydro-Electric  Machine. — It  was  discovered  in 
1840  that  a  steam-boiler  electrically  insulated  gave  out  sparks, 
and  that  the  steam  issuing  from  it  was  also  electrified.    Hence  re- 
sulted the  construction  of  the  hydro-electric  machine.    It  consists 
of  a  boiler  mounted  on  glass  pillars,  and  furnished  with  a  row  of 


THE    QUADRANT    ELECTROMETER  249 

jet-pipes  and  a  metallic  plate,  against  which  the  steam  strikes. 
The  prime  conductor,  to  which  the  steam-plate  is  attached,  is 
electrified  positively,  and  the  boiler  itself  negatively.  Professor 
Faraday  ascertained  that  the  electricity  in  this  case  is  developed, 
not  by  evaporation  or  condensation,  but  by  the  friction  of  watery 
particles  in  the  jet-pipes.  That  the  machine  may  act  with  energy, 
it  was  found  necessary  to  make  the  interior  of  the  jet-pipes  angu- 
lar, and  quite  irregular. 

In  connection  with  the  subject  of  induction  will  be  described 
a  machine  of  still  more  recent  invention,  and  known  as  the  induc- 
tion machine. 

400.  The  Quadrant  Electrometer. — In  order  to  measure 
the  intensity  of  electricity  in  the  prime  con- 
ductor, there  is  set  upon  it,  whenever  desired,         _FlG' 

a  quadrant  electrometer  (Fig.  228).  This  con- 
sists of  a  pillar,  d,  about  six  inches  high,  having 
a  graduated  semicircle,  c,  attached  to  one  side, 
and  a  delicate  rod  and  ball,  a,  suspended  from 
the  centre  of  the  semicircle.  As  the  conductor 
becomes  electrified,  the  rod  is  repelled  from  the 
pillar,  and  the  arc  passed  over  indicates  rudely 
the  degree  of  electrical  intensity. 

401.  First   Phenomena    cf  the    Ma- 
chine.— When  an  electrical  machine  is  skill- 
fully fitted  up,  and  works  well,  there  is  first 

perceived,  on  turning  it,  a  crackling  sound;  and  then,  on  bring- 
ing the  knuckles  toward  the  prime  conductor,  a  brilliant  spark 
leaps  across,  causing  a  sharp  pricking  sensation.  If  the  room  be 
darkened,  brushes  of  pale  light  are  seen  to  dart  off  continually 
from  the  most  slender  parts  of  the  prime  conductor,  with  a  hiss- 
ing or  fluttering  noise,  while  circles  of  light  snap  along  the  glass 
between  the  rubbers  and  the  edges  of  the  covers.  When  electricity 
is  escaping  plentifully  from  the  machine,  a  person  standing  near 
also  perceives  a  peculiar  odor,  which  is  that  of  ozone,  and  which 
seems  always  to  accompany  the  development  of  electricity. 

Therefore,  at  least  four  of  the  senses  are  directly  affected  by 
this  remarkable  agency,  while  magnetism  affects  none  of  them. 

The  phenomena  of  repulsion  of  like  and  attraction  of  unlike 
electricities,  are  well  shown  by  the  machine.  A  skein  of  thread 
or  a  tuft  of  hair,  suspended  from  the  prime  conductor,  will,  as 
soon  as^the  plate  is  revolved,  spread  into  as  wide  a  space  as  possi- 
ble, by  the  repellency  of  the  fibers  which  are  electrified  alike. 
Melted  sealing-wax  is  thrown  off  in  fine  threads,  and  dropping 


250 


STATICAL    ELECTRICITY. 


water  is  diverged  into  delicate  filaments.  Even  air,  on  those  parts 
of  the  prime  conductor  which  are  most  strongly  charged,  becomes 
so  self-repellent  as  to  fly  off  in  a  stream  of  wind,  which  is  plainly 
felt. 

On  the  other  hand,  light  bodies,  when  brought  toward  the  ma- 
chine while  in  action,  instantly  fly  to  the  prime  conductor ;  for 
that  is  positive,  but  the  nearer  sides  of  the  other  bodies  are  made 
negative  by  induction. 

The  difference  between  substances  as  to  their  conducting  qual- 
ity is  readily  perceived  by  setting  the  quadrant  electrometer  on 
the  prime  conductor,  raising  the  index  by  turning  the  plate,  and 
then  touching  the  prime  conductor  with  the  remote  end  of  the 
body  to  be  tried.  If  an  iron  rod,  or  even  a  fine  iron  wire,  be  thus 
applied,  the  index  will  fall  instantly;  a  long  dry  wooden  rod 
will  cause  it  to  descend  slowly,  while  a  glass  rod  will  produce  no 
effect  at  all.  These  experiments  show  that  iron  is  a  perfect  con- 
ductor, wood  an  imperfect  conductor,  and  glass  a  non-conductor. 

402.  Coulomb's  Torsion  Balance. — "When  a  long  fine  wire 
is  stretched  by  a  small  weight,  its  elasticity  of  torsion  is  a  very 
delicate  force,  which  is  successfully  employed  for  the  measurement 
of  other  small  forces.    When  such  a  wire  is  twisted  through  differ- 
ent angles,  the  force  of  torsion  is  found  to  vary  as  the  angle  of  tor- 
sion ;  it  is  therefore  easy  to  measure  the 

force  which  is  in  equilibrium  with  tor- 
sion. The  torsion  balance  is  represented 
in  Fig.  229.  The  needle  of  lac,  n  o,  is 
suspended  by  a  very  fine  wire  from  a 
stem  at  the  top  of  the  tube  d.  The  cap 
of  the  tube,  e,  is  a  graduated  circle, 
whose  exact  position  is  marked  by  the 
index,  a.  The  stem  from  which  the 
wire  hangs  is  held  in  place  in  the  centre 
of  the  cap  by  friction,  but  can  be  turned 
round  so  as  to  place  the  needle  in  any 
direction  desired.  At  the  end  of  the  lac 
needle  is  a  small  disk  of  brass-leaf,  n, 
and  by  its  side  a  gilt  ball,  m,  connected 
with  the  handle,  r,  by  the  glass  rod,  i. 
This  apparatus  is  suspended  in  the  glass 
cylinder,  covered  with  a  glass  plate,  on 
the  centre  of  which  the  tube  d  is  fastened.  There  is  a  graduated 
circle  around  the  C}dinder  on  the  level  of  the  needle. 

403.  Law  of  Electrical  Force  as  to  Distance. — Adjust- 
ment is  now  made  by  turning  the  stem  so  that,  while  the  wire  is 


FIG.  229. 


LAW    OF    ELECTRICAL    FORCE.  251 

in  its  natural  condition,  the  disk,  n,  touches  the  ball,  m,  and  is  at 
zero,  and  the  index  at  top  also  at  zero  on  the  circle  e.  Let  a  mi- 
nute charge  of  electricity  be  communicated  to  m,  and"  it  will  repel 
n,  and  cause  it,  after  a  few  oscillations,  to  settle  at  a  certain  dis- 
tance— suppose,  for  instance,  at  36°.  The  circle  e  is  now  turned  in 
the  opposite  direction,  until  the  needle  is  brought  within  18°  of 
the  ball  m.  In  order  to  bring  it  thus  near,  the  index  has  to  be 
turned  126°,  which  added  to  the  18°,  makes  the  whole  torsion 
144°,  o-rfour  times  as  great  as  before.  Therefore,  at  one-half  the 
distance  there  is  four  times  the  repulsion.  In  like  manner,  it  is 
found  that  at  one-third  the  distance  there  is  nine  times  the  repul- 
sion. Hence,  the  law, 

Electrical  repulsion  varies  inversely  as  the  square  of  the  dis- 
tance. 

In  a  manner  somewhat  similar  to  the  foregoing,  it  was  conclu- 
sively proved  by  Coulomb  that  electrical  attraction  obeys  the  same 
law  of  distance,  though  there  is  more  practical  difficulty  in  per- 
forming the  experiments.  But  if  the  electrified  body  m  is  placed 
outside  of  the  circle  described  by  n,  so  that  the  latter  is  allowed  to 
vibrate  both  to  the  right  and  left,  the  square  of  the  number  of 
vibrations  in  a  given  time  becomes  a  measure  of  the  attractive 
force,  as  in  the  case  of  the  pendulum  (Art.  170). 

404.  Waste  of  Electricity  from  an  Insulated  Body.— 

In  making  accurate  investigations  like  the  foregoing,  in  which 
considerable  time  is  necessarily  occupied,  a  difficulty  arises  from 
the  loss  of  the  electrical  charge.  The  first  and  most  obvious 
source  of  waste  is  the  moisture  in  the  air,  which  conducts  away 
the  fluid ;  but  this  may  be  nearly  avoided  by  setting  into  the  cyl- 
inder a  cup  of  dry  lime,  or  other  powerful  absorbent  of  moisture, 
as  represented  in  the  figure.  A  second  is  the  imperfect  insulation 
afforded  by  even  the  most  perfect  non-conductors.  A  third  is  the 
mobility  of  the  air,  whose  particles,  when  they  have  touched  the 
electrified  body,  and  become  charged,  are  repelled,  taking  away 
with  them  the  charge  they  have  received.  The  loss  in  these  ways 
is  very  slight,  when  the  charge  is  small,  and  allowance  can  be 
made  for  it  with  a  good  degree  of  accuracy.  But  when  bodies  are 
highly  charged,  they  lose  their  electricity  at  a  rapid  rate. 

405.  An  Electrical  Charge  Lies  at  the  Surface. — This 
is  proved  in  many  ways.    A  hollow  ball,  no  matter  how  thin,  will 
receive  as  large  a  quantity  of  electricity  as  a  solid  one.    Hence  it 
is  that  the  prime  conductor  of  the  electrical  machine,  and  metallic 
articles  of  electrical  apparatus  generally,  are  made  of  sheet  brass, 
for  the  sake  of  lightness. 

Let  a  metallic  ball,  supported  on  a  glass  pillar,  be  charged 


252  STATICAL    ELECTEICITY. 

with  either  kind  of  electricity.  Then  apply  to  it  two  thin  metal- 
lic hemispheres,  by  means  of  insulating  handles.  If  they  now  be 
quickly  removed  from  the  ball,  all  the  electricity  which  was  pre- 
viously on  the  ball  is  found  On  the  hemispheres. 

Let  a  dish,  a  (Fig.  230),  be  made  of  two  brass  rings  and  cam- 
bric sides  and  bottom,  with  an  insulating 
handle,  #,  attached  to  the  larger  ring.    If 
this  vessel  be  charged  with  electricity,  the 
charge  is  found  on  the  outside ;  turn  it 
over  quickly,  so  as  to  throw  it  the  other 
side  out,  and  the  charge  is  instantly  found 
on  the  outside  again,  and  none  on  the  inside.    It  may  be  inverted 
several  times  with  the  same  result,  before  the  charge  becomes  too 
feeble  to  be  perceived. 

If  cavities  are  sunk  into  a  solid  conductor,  no  sensible  quantity 
of  electricity  is  found  at  the  bottom  of  such  cavities.  In  experi- 
ments of  this  kind,  Coulomb  found  his  torsion  balance  (Fig.  229) 
of  great  service.  A  proof  plane,  as  he  termed  it — that  is,  a  small 
piece  of  gilt  paper  cemented  upon  the  end  of  a  slender  rod  of  lac, 
was  first  touched  to  that  part  of  an  electrified  body  which  was  to 
be  examined,  and  then  applied  to  the  ball  of  the  instrument.  The 
distance  to  which  the  needle  was  repelled  indicated  the  intensity 
of  electricity  at  the  point  in  question.  The  charge  taken  from  the 
bottom  of  an  abrupt  cavity  was  never  sufficient  to  move  the 
needle. 

Another  proof  that  the  charge  occupies  only  the  outside  sur- 
face is  that  the  intensity  diminishes  as  the  surface  is  enlarged, 
while  the  mass  of  the  conductor  remains  the  same.  A  metallic 
ribbon  rolled  upon  an  insulated  cylinder  may  be  unrolled,  and 
thus  the  surface  enlarged  to  any  extent.  An  electroscope  standing 
on  the  instrument  will  fall  as  the  ribbon  is  unrolled,  and  rise 
when  it  is  again  rolled  up. 

406.  Distribution  of  a  Charge  on  the  Surface. — Devel- 
oped electricity  resides  at  the  surface  of  a  body,  as  we  have  seen, 
but  is  not  uniformly  diffused  over  it,  except  in  the  case  of  the 
sphere.  In  general,  the  more  prominent  the  part,  and  the  more 
rapid  its  curvature,  the  more  intensely  is  the  fluid  accumulated 
there. 

In  a  long  slender  rod,  nearly  the  whole  charge  is  collected  at 
the  extremities.  On  the  surface  of  an  ellipsoid  it  is  found  to  be 
arranged  according  to  a  very  simple  law,  namely :  the  quantity  of 
the  charge  at  each  point  varies  as  the  diameter  through  that  point. 
But  the  tendency  to  escape  increases  at  a  more  rapid  rate,  and 
varies  as  the  square  of  the  diameter.  Hence  it  is  that  electricity 


CHARGE  HELD  ON  THE  SURFACE.      353 

is  so  rapidly  dissipated  from  points,  which  may  be  regarded  as  the 
extremities  of  ellipsoids  indefinitely  elongated.  If  the  surface  of  a 
body  is  partly  convex  and  partly  concave,  the  distribution  is  still 
more  unequal;  nearly  all  the  charge  collects  on  the  convex  parts; 
and  if  the  concavities  are  deep  or  abrupt,  like  those  mentioned  in 
Art.  405,  no  sign  of  electricity  is  discovered  in  them. 

407.  Rotation  by  Unbalanced  Pressure.— As  the  electric 
charge  on  the  surface  of  a  body  presses  outward  in  all  directions, 
wherever  it  escapes  from  a  point,  there  the  pressure  is  removed; 
consequently,  on  the  opposite  part  there  is  unbalanced  pressure. 
Therefore,  if  the  body  is  delicately  suspended,  and  one  or  more 
points   are  directed   tangentially,  the  unbalanced  pressure  will 
cause  rotation  in  the  opposite  direction,  just  as  Barker's  mill  ro- 
tates by  the  unbalanced  pressure  of  water.    Electrical  wheels  and 
orreries  are  revolved  in  this  way. 

A  windmill  may  also  be  revolved  by  the  stream  of  air  issuing 
from  a  stationary  point  attached  to  the  prime  conductor  (Art.  401). 

408.  The  Charge  Hsfd  on  the  Surface  by  Atmospheric 
Pressure. — The  mutual  repellency,  which  drives  the  particles 
asunder  till  they  reach  the  surface  of  the  conductor,  tends  to  make 
them  escape  in  all  directions  from  that  surface ;  and  it  is  the  air 
alone  which  prevents.    For  if  one  extremity  of  a  charged  and  in- 
sulated conductor  extends  into  the  receiver  of  an  air-pump,  the 
charge  is  dissipated  by  degrees,  as  the  receiver  is  exhausted ;  and 
when  the  exhaustion  is  as  complete  as  possible,  the  mgst  abundant 
supply  from  the  machine  fails  to  charge  the  conductor.    As  the 
atmospheric  pressure  is  limited  to  about  15  Ibs.  per  square  inch,  so 
the  amount  of  charge  is  limited  which  can  be  retained  on  a  con- 
ductor of  given  form.    Hence  the  reason  for  the  well-known  fact 
that  the  prime  conductor  receives  all  the  charge  which  it  is  capa- 
ble of  retaining  in  one  or  two  turns  of  the  machine.    All  that  is 
gained  over  and  above  this,  by  continuing  to  turn,  flies  off  through 
the  air. 


CHAPTER    III. 

ELECTRICITY  BY  INDUCTION.— LEYDEN  JAR. 

409.  Elementary  Experiment,— When  an  electrified  body 
is  placed  near  one  which  is  unelectrified,  but  not  within  the 
sphere  of  communication,  the  natural  electricities  of  the  latter  are 


254 


STATICAL    ELECTRICITY. 


decomposed,  one  being  attracted  toward  the  former,  the  other  re- 
pelled from  it  (Art.  396).  Thus  the  ends  become  electrified  by 
the  influence  of  the  first  body,  without  receiving  any  electricity 
from  it.  Let  A  (Fig.  231)  be  charged  with  positive  electricity, 


FIG.  231. 


A  A  1  A  A 


and  let  the  insulated  conductor,  B  C,  be  furnished  with  several 
electroscopes,  as  represented.  Those  nearest  the  ends  will  diverge 
most,  and  the  others  less  according  as  they  are  nearer  the  centre, 
where  there  is  no  sign  of  electricity.  By  taking  off  small  quanti- 
ties with  the  proof-plane,  and  testing  them,  it  is  found  that  nega- 
tive electricity  occupies  the  end  nearest  to  A,  and  positive  the 
remote  end.  Remove  the  bodies  to  a  distance  from  each  other, 
and  B  C  returns  to  its  unelectrified  condition ;  bring  them  near 
again,  and  it  is  electrified  as  before.  As  this  electrical  state  is  in- 
duced upon  the  conductor  by  the  electrified  body  in  its  vicinity, 
without  any  communication  of  electricity,  it  is  said  to  be  electri- 
fied by  induction.  If  A  is  first  charged  with  the  negative  elec- 
tricity, the  two  electricities  of  B  C  will  be  arranged  in  reversed 
order ;  the  positive  will  be  attracted  to  the  nearest  end,  the  nega- 
tive repelled  to  the  farthest. 

Electrical  induction  is  exactly  analogous  to  magnetic  induc- 
tion ;  the  opposite  kind  is  developed  at  the  nearer  end,  and  the 
like  kind  at  the  remote  end. 

410.  Successive  Actions  and  Reactions.— If  A  is  itself 
an  insulated  conductor,  the  foregoing  is  not  the  entire  effect ;  for 
a  reflex  influence  is  exerted  by  the  electricity  in  the  nearer  end  of 
the  conductor.  Let  A  have  a  positive  charge,  as  at  first.  After 
the  negative  electricity  is  attracted  to  the  nearer  end  of  B  C,  it  in 
turn  attracts  the  positive  charge  of  A,  and  accumulates  it  on  the 
nearest  side,  leaving  the  remote  side  less  strongly  charged  than 
before.  This  is  shown  by  electroscopes  attached  to  the  opposite 
sides  of  A.  The  charge  of  A,  being  now  nearer,  will  exert  more 
power  on  B  C,  separating  more  of  its  original  electricities,  and 


INDUCTION.  255 

thus  making  the  nearest  end  more  strongly  negative  and  the  re- 
mote end  more  strongly  positive  than  before ;  and  this  new  ar- 
rangement of  fluids  in  B  C  causes  a  second  reaction  upon  A,  of 
the  same  kind  as  the  first.  Thus  an  indefinite  diminishing  series 
of  adjustments  takes  place  in  a  single  moment  of  time. 

411.  Division  of  the  Conductor. — Suppose  that  before  the 
experiment  begins,  B  C  is  in  two  parts  with  ends  in  contact;  the 
entire  series  of  mutual  actions  takes  place  as  already  described. 
Now,  while  A  remains  in  the  vicinity,  let  the  parts  of  B  (7  be  sep- 
arated ;  then  the  negative  electricity  is  secured  in  the  nearest  half, 
and  the  positive  in  the  other.    And  if  A  is  now  removed,  the  pos- 
itive charge  is  diffused  over  the  more  distant  half.     Thus  each 
kind  of  electricity  can  be  completely  separated  from  the  other  by 
means  of  induction. 

Here  we  find  a  marked  difference  between  magnetism  and  fric- 
tional  electricity.  The  electricities  may  be  secured  in  their  sep- 
arate state,  one  in  one  conductor,  the  other  in  another.  In  mag- 
netism this  is  not  possible ;  for  when  an  iron  bar  is  magnetized, 
and  then  broken,  each  kind  of  magnetism  is  found  in  each  half  of 
the  bar.  At  the  point  of  division  both  polarities  exist,  and  as  soon 
as  the  bar  is  broken,  they  manifest  themselves  there  as  strongly  as 
at  the  extremities. 

412.  Effect  of  Lengthening  the  Conductor. — If  the  con- 
ductor, B  C,  is  lengthened,  the  accumulation  on  the  adjacent  parts 
of  the  two  bodies  is  somewhat  increased.     The  positive  electricity 
which,  at  the  remote  end  of  the  shorter  conductor,  operated  in  some 
degree  by  its  repulsion  to  prevent  accumulation  on  the  nearest 
side  of  A,  is  now  driven  to  a  greater  distance ;  and  therefore  a 
larger  charge  will  come  from  the  remote  to  the  nearer  side  of  A, 
which  in  turn  attracts  more  negative  to  the  nearer  end  of  B  C, 
and  thus  a  new  series  of  actions  and  reactions  takes  place  in  addi- 
tion to  the  former.    To  obtain  the  greatest  effect  from  this  cause, 
the  conductor,  B  C,  is  connected  with  the  earth — that  is,  it  is  un- 
insulated; then  the  positive  part  of  its  decomposed  electricities  is 
driven  to  the  earth,  and  entirely  disappears,  and  the  negative  part 
is  attracted  to  the  nearer  end ;  so  that,  when  the  series  of  adjust- 
ments is  completed,  the  remote  end  of  the  conductor  is  in  the 
neutral  state.    This  experiment  is  performed  by  touching  the 
finger  to  the  conductor,  after  it  has  become  electrified  by  induc- 
tion.   The  electroscope  nearest  to  A  instantly  rises  a  little  higher, 
and  the  distant  ones  collapse. 

If  the  original  charge  in  A  was  negative  instead  of  positive,  the 
foregoing  experiments  are  in  all  particulars  the  same,  except  that 
the  order  of  the  two  fluids  is  reversed. 


256  STATICAL    ELECTRICITY. 

413.  Disguised  Electricity. — The  electricity  which  occu- 
pies the  surface  of  the  prime  conductor,  or  any  other  body  electri- 
fied in  the  ordinary  way,  and  which  is  kept  from  diffusing  itself 
in  every  direction  only  by  the  pressure  of  the  air  (Art.  408),  is 
called  free  electricity;  for  it  will  instantly  spread  over  the  surface 
of  other  conductors,  when  they  are  presented,  and  therefore  will 
be  lost  in  the  earth,  the  moment  a  communication  is  made.    But 
the  electricity  which  is  accumulated  by  the  inductive  influence  is 
not  free  to  diffuse  itself;  the  same  attractive  force  which  has  con- 
densed it  still  holds  it  as  near  as  possible  to  the  original  charge ; 
and  if  we  touch  the  electrified  body  with  the  hand,  the  electricity 
does  not  pass  off;  it  is  therefore  called  disguised  electricity.    In 
this  respect  the  two  fluids  on  the  contiguous  sides  of  A  and  B  G 
are  alike ;  either  may  be  touched,  or  in  any  way  connected  with 
the  earth,  but,  unless  communication  is  made  between  them,  or 
unless  they  are  both  allowed  to  pass  to  the  earth,  they  hold  each 
other  in  place  by  their  mutual  attraction,  and  show  none  of  the 
phenomena  of  free  electricity. 

414.  A  Series  of  Conductors.— If  another  insulated  con- 
ductor, /),  is  placed  near  to  the  remote  end  of  B  C,  and  A  is 
charged  positively,  then  that  extremity  of  B  C  nearest  to  D  is  in- 
ductively charged  with  positive,  as  already  stated.    Hence,  the 
electricities  of  D  are  separated,  the  negative  approaching  B  C,  and 
the  positive  withdrawing  from  it ;  there  is  therefore  the  same  ar- 
rangement of  fluids  in  both  bodies,  but  a  less  intensity  in  D  than 
in  B  G.    For,  on  account  of  distance,  the  positive  is  not  so  in- 
tensely accumulated  at  the  remote  end  of  B  C  as  in  the  original 
body  A,  and  therefore  a  less  force  operates  on  D  than  on  B  G. 
The  same  effects  are  produced  in  a  less  and  less  degree  in  an  in- 
definite series  of  bodies ;  and  the  shorter  they  are,  the  more  nearly 
equal  will  be  the  successive  accumulations.    The  same  facts  were 
noticed  in  a  series  of  magnets. 

415.  An  Electrified   Body  Attracts   an  Unelectrified 
Body. — This  fact,  which  is  the  first  to  be  noticed  in  observing 
electrical  phenomena  (Art.  401),  is  explained  by  induction.    If 
B  C  is  light,  and  delicately  suspended,  a  consequence  of  the  ar- 
rangement of  fluids  already  described  is,  that  B  C  will  move  to- 
ward A.    For,  according  to  the  law  of  distance   (Art.  403),  the 
negative  in  the  nearer  part  is  attracted  more  strongly  than  the 
positive  in  the  remote  part  is  repelled ;  hence  the  body  yields  to 
the  greater  force,  and  moves  toward  A.    That  the  attracted  fluid 
does  not  leave  the  body,  B  G,  behind,  and  go  to  A,  is  owing  to  the 
fact,  noticed  in  Art.  408,  that  the  body  and  the  electricity  are  con- 
fined to  each  other  by  atmospheric  pressure. 


THE    LEYDEN    JAR. 


257 


FIG.  232. 


416.  The  Inductive  Action  Greatly  Increased. — In  the 

experiments  as  now  described,  the  inductive  influence  is  feeble, 
and  the  accumulation  of  electricities  very  small ;  for  the  bodies 
present  toward  each  other  only  a  limited  extent  of  area,  and  they 
are  necessarily  as  much  as  four  or  live  inches  distant,  in  order  to 
prevent  the  fluid  from  passing  across.  By  giving  the  bodies  such 
a  form  that  a  large  extent  of  surface  may  be  equidistant,  and  then 
interposing  a  solid  non-conductor,  as  glass,  between  them,  so  that 
the  distance  may  be  reduced  to  one-eighth  of  an  inch  or  less,  it  is 
easy  to  increase  the  attracting  and  repelling  forces  many  thousands 
of  times.  Let  a  glass  plate,  C  D  (Fig.  232),  supported  on  a  base, 
have  attached  to  the  middle  of  each 
side  a  rectangular  piece  of  tin-foil. 
This  is  called  a  Franklin  plate.  Let 
A  be  connected  with  one  coating,  and 
B  with  the  other.  If,  now,  A  forms 
a  part  of  the  prime  conductor  of  an 
electrical  machine,  and  B  has  com- 
munication with  the  earth,  we  are 
prepared  to  notice  the  remarkable 
phenomena  of  the  Leyden  jar.  If 
the  amount  of  surface  and  the  thick- 
ness of  glass  are  the  same,  the  partic- 
ular form  of  the  instrument  is  im- 
material; but,  for  most  purposes,  a 
vessel  or  jar  is  more  convenient  than  a  pane  of  glass  of  equal  sur- 
face, and  is  generally  employed  for  electrical  experiments. 

417.  The  Leyden  Jar. — This  article  of  electrical  apparatus 
consists  of  a  glass  jar  (Fig.  233),  coated  011  both  sides  with  tin-foil, 
except  a  breadth  of  two  or  three  inches  near  the  top, 

which  is  sometimes  varnished  for  more  perfect  insula- 
tion. Through  the  cork  passes  a  brass  rod,  which  is  in 
metallic  contact  with  the  inner  coating,  and  terminates 
in  a  ball  at  the  top. 

On  presenting  the  knob  of  the  jar  near  to  the  prime 
conductor  of  an  electrical  machine,  while  the  latter  is 
in  operation,  a  series  of  sparks  passes  between  the  con- 
ductor and  the  jar,  which  will  gradually  grow  more  and 
more  feeble,  until  they  cease  altogether.  The  jar  is  then 
said  to  be  charged.  If  now  we  take  the  diseharging-rod, 
which  is  a  curved  wire,  terminated  at  each  end  with  a 
knob,  and  insulated  by  glass  handles  (Fig.  234),  and  apply  one 
of  the  knobs  to  the  outer  coating  of  the  jar,  and  bring  the  other 
to  the  knob  of  the  jar,  a  flash  of  intense  brightness,  accompanied 
17 


Fio.  233. 


258  STATICAL    ELECTRICITY. 

by  a  sharp  report,  immediately  ensues.    This  is  the  discharge  of 
the  jar. 

If,  instead  of  the  discharging-rod,  FIG.  234. 

a  person  applies  one  hand  to  the  out- 
side of  the  charged  jar,  and  brings  the 
other  to  the  knob,  a  sudden  shock  is 
felt,  convulsing  the  arms,  and  when 
the  charge  is  heavy,  causing  pain 
through  the  body.  The  shock  pro- 
duced by  electricity  was  first  discov- 
ered accidentally  by  persons  experi- 
menting with  a  charged  phial  of  water.  This  occurred  in  Leyden, 
and  led  to  the  construction  and  name  of  the  Leyden  jar. 

418.  Theory  of  the  Leyden  Jar. — This  instrument  accu- 
mulates and  condenses  great  quantities  of  electricity  on  its  sur- 
faces, upon  the  principle  of  mutual  attraction  between  unlike 
electricities,  one  of  which  is  furnished  by  the  machine,  the  other 
obtained  from  the  earth  by  induction.    First,  suppose  the  outer 
coating  insulated;  a  spark  of  the  positive  electricity  passes  from 
the  prime  conductor  to  the  inner  coating,  which  tends  to  repel 
the  positive  from  the  outer  coating ;  but  as  the  latter  cannot  es- 
cape, it  remains  to  prevent,  by  its  counter-repulsion,  any  addition 
to  the  charge  of  the  inside,  and  thus  the  process  stops.    But  now 
connect  the  outer  coating  with  the  earth,  and  immediately  some 
of  its  positive  electricity,  repelled  by  the  charge  on  the  inside, 
passes  off,  while  its  negative  is  attracted  close  upon  the  glass,  and 
gives  room  for  the  accession  of  more  from  the  earth.     The  slight 
condensation  of  negative  upon  the  outside,  by  its  attraction,  con- 
denses the  positive  of  the  inner  coating,  and  allows  a  second  spark 
to  pass  in  from  the  prime  conductor.    This  produces  the  same 
effect  as  the  first,  and  a  second  addition  of  negative  is  made  to  the 
outer  coating,  the  latter  being  obtained  from  the  earth  as  before. 
These  actions  and  reactions  go  on  in  a  diminishing  series,  till 
there  is  a  great  accumulation  of  the  two  electricities,  held  by  mu- 
tual attraction  as  near  each  other  as  possible,  on  opposite  sides  of 
the  glass.     The  jar  in  this  condition  is  said  to  be  charged. 

If  the  positive  electricity  is  on  the  inner  coating,  the  jar  is 
said  to  be  positively  charged  ;  if  on  the  outside,  negatively  charged. 

419.  The  Spontaneous  Discharge. — This  occurs  when  the 
quantities  accumulated  are  so  great  that  their  attraction  will  cause 
them  to  fly  together  with  a  flash  and  report  over  the  edge  of  the 
jar.    If  the  glass  is  soiled  or  damp,  the  fluids  may  pass  over  and 
mingle  with  only  a  hissing  noise,  in  which  case  it  is  impossible  for 
the  jar  to  be  highly  charged. 


SERIES    OF    JARS.  259 

If  the  glass  is  clean  and  dry,  and  especially  if  varnished  with 
gum  lac,  a  charge  may  not  wholly  disappear  for  days,  or  ever, 
weeks. 

420.  Series  of  Jars. — The  same  amount  of  electricity  from 
the  prime  conductor  which  is  required  to  charge  one  jar  will 
charge  an  indefinite  series,  the  strength  of  the  charge  being  less  and 
less  from  the  first  to  the  last.    This  case  is  analogous  to  the  series 
of  conductors  (Art.  414).     Insulate  a  series  of  jars,  A,  B,  C,  &c., 
and  connect  the  inner  coating  of  A  with  the  prime  conductor,  and 
its  outer  coating  with  the  inner  coating  of  B,  the  outer  of  B  with 
the  inner  of  6',  and  so  on.     Then,  as  A  is  charged,  the  positive 
electricity  of  its  outer  coating,  instead  of  passing  to  the  earth,  goes 
to  the  inside  of  B,  and  that  on  the  outside  of  B  to  the  inside  of 
(7,  &c.,  while  that  on  the  outside  of  the  last  in  the  series  passes  to 
the  earth.     Thus  each  jar  is  charged  positively  by  the  inductive 
influence  of  the  preceding,  just  as  a  series  of  magnets  is  formed 
with  poles  in  the  same  order  by  a  succession  of  magnetic  induc- 
tions. 

421.  Division  of  a  Charge  in  any  Given  Ratio. — If  one 

of  two  jars  be  charged,  and  the  other  not,  and  if  the  inner  coat- 
ings be  brought  into  communication,  and  also  the  outer  coatings, 
the  charge  of  the  first  jar  is  instantly  diffused  over  the  two,  with 
a  report  like  that  of  a  discharge.  In  this  way  a  charge  may  be 
halved,  or  divided  in  any  other  ratio,  according  to  the  relative  sur- 
faces of  the  jars. 

The  self-repellency  of  each  fluid  tends  to  diffuse  it  over  a 
greater  surface,  and  they  will  be  thus  diffused  if  allowed  to  remain 
within  each  other's  attracting  influence ;  but  one  of  the  fluids  will 
not  be  spread  over  the  coatings  of  another  jar,  unless  opportunity 
is  given  for  both  to  do  it. 

An  experiment  somewhat  resembling  the  foregoing  is  this: 
charge  two  equal  jars,  one  positively,  the  other  negatively,  and  in- 
sulate them  both.  If  the  two  knobs  be  connected  by  a  conductor, 
the  electricities,  notwithstanding  their  strong  attraction,  will  not 
unite ;  for  each  is  held  disguised  by  that  on  the  other  side  of  the 
glass.  But  if  the  outer  coatings  are  first  connected,  then,  on  join- 
ing the  knobs,  the  jars  are  both  discharged  at  once. 

422.  Use  of  the  Coatings.— If  a  jar  is  made  with  a  wide 
open  top,  and  the  coatings  movable,  then,  after  charging  the  jar 
and  removing  the  coatings,  very  little  of  the  electricities  adheres 
to  the  latter,  but  nearly  the  whole  remains  on  the  glass.    The 
same  mutual  attraction  which  condensed  them  at  first  still  holds 
them  there  after  the  coatings  are  removed.    When  they  come  to 


260  STATICAL    ELECTRICITY. 

be  replaced,  the  jar  can  be  discharged  as  usual.  But  the  coatings 
are  necessary  in  charging,  to  diffuse  the  electricity  over  those  parts 
of  the  glass  which  they  cover,  and  also  in  discharging,  to  conduct 
off  the  whole  charge  at  once. 

423.  The  Free  Portion  of  an  Electrical  Charge.— Either 
kind  of  electricity  is  said  to  be  free  when  it  remains  on  a  body 
only  because  held  by  the  pressure  of  the  air;  but  if  held  by  the 
attraction  of  the  opposite  kind,  it  is  said  to  be  disguised  (Art.  413). 
Nearly  all  the  electricity  of  a  charged  jar  is  disguised,  but  not  the 
whole. 

The  moment  after  a  jar  is  charged  there  is  a  small  quantity  of 
free  electricity  on  the  coating  to  which  the  fluid  was  furnished  in 
charging,  but  not  on  the  other.  But  after  the  jar  has  stood 
charged  some  minutes,  a  little  is  free  on  both  coatings.  If  the 
charged  jar  be  upon  an  insulating  stand,  and  the  finger  brought 
to  one  coating,  a  slight  spark  is  taken  off ;  if  it  be  touched  again 
immediately,  there  is  no  spark,  for  the  free  electricity  all  escaped 
by  the  first'  contact.  Let  the  finger  now  be  brought  to  the  other 
coating,  and  a  spark  flies  from  that.  Immediately  afterward  a 
second  spark  can  be  taken  from  the  first  coating,  and  so  on  alter- 
nately for  hundreds  of  times  usually  before  the  charge  wholly  dis- 
appears. What  is  removed  at  each  contact  is  the  free  part  of  the 
charge,  which  always  appears  alternately  on  the  two  coatings.  If 
a  small  electroscope  be  connected  with  each  coating,  the  fluid  al- 
ternately set  free  is  indicated  to  the  sight.  The  electroscope  on 
the  coating  which  is  touched  instantly  falls,  and  the  other  rises. 

424.  Explanation   of   this  Phenomenon. — The  positive 
electricity  which  is  conveyed  to  the  inner  coating,  in  charging  a 
jar,  attracts  to  the  outer  coating  from  the  earth  a  quantity  of  the 
negative  fluid  which  is  a  little  less  than  itself.    This  is  because  of 
the  thickness  of  the  glass.    If  it  were  infinitely  thin,  the  negative 
would  be  just  equal  to  the  positive,  and  they  would  neutralize 
each  other,  and  both  be  perfectly  disguised.     But  as  the  glass  has 
some  thickness,  the  positive  exceeds  the  negative,  and  disguises  it. 
Now  if  the  jar,  after  being  charged,  is  insulated,  it  is  obvious  that 
the  negative  charge  on  the  outer  coating  cannot  disguise  all  the 
positive  (which  is  more  than  itself),  but  only  a  quantity  a  little 
less  than  itself.     Hence  there  must  be  a  little  of  the  positive  on 
the  inner  coating  in  a  free  state.    By  touching  the  knob,  we  allow 
this  free  portion  to  pass  off,  and  there  is  left  less  of  the  positive  in 
the  inner  coating  than  there  is  of  the  negative  in   the  outer. 
Therefore,  all  the  negative  cannot  now  be  disguised,  but  a  slight 
quantity  is  liberated  and  ready  to  pass  off  as  soon  as  touched. 
And  thus,  by  alternate  contacts,  the  process  of  discharge  goes  on, 


VIBRATIONS    AND    REVOLUTIONS.  261 

the  series  being  longer  as  the  glass  is  thinner,  because  then  the 
two  quantities  are  more  nearly  equal. 

425.  Electrical   Vibrations   and   Revolutions. — If  two 

jars  be  charged  in  opposite  ways,  and  a  figure  made  of  pith  be  sus- 
pended  between  the  knobs  by  a  long  thread,  it  will  be  attracted  by 
that  knob  whose  action  on  it  happens  to  be  greatest.  As  soon  as 
it  touches,  it  is  charged  with  that  kind  and  repelled,  and  of  course 
attracted  by  the  other  knob,  which  is  in  the  opposite  state ;  thus 
it  vibrates  between  them,  causing  a  very  slow  discharge  of  both 
jars.  In  this  case,  the  outside  of  the  jars  not  being  insulated,  the 
electricity,  which  is  slowly  set  free  on  the  outside,  passes  off,  and 
therefore  there  is  always  some  free  electricity  on  the  knob  to  be 
imparted  to  the  vibrating  figure. 

The  electricity  of  the  prime  conductor  will  also  cause  vibra- 
tions, without  the  use  of  a  jar.  Suspend  from  it  a  metallic  disk 
horizontally  a  few  inches  above  another  which  is  connected  with 
the  earth ;  then  if  a  glass  cylinder  surround  the  two  disks  so  as  to 
prevent  escape,  a  number  of  pith  balls  between  the  disks  will  con- 
tinue to  vibrate  up  and  down  so  long  as  the  machine  is  in  action. 
Each  ball  lying  on  the  lower  disk,  being  electrified  by  induction 
in  the  opposite  way  from  the  upper  one,  springs  up  to  it,  and 
then,  being  charged  in  the  same  way,  is  repelled. 

In  a  similar  manner  a  chime  of  bells  may  be  rung,  orreries  re- 
volved, &c. 

426.  Residuary  Charge. — If  a  jar  stand  charged  a  few  min- 
utes, and  after  the  discharge  remain  some  minutes  more,  then  a 
second,  and  possibly  a  third,  discharge  can  be  made;  but  these  are 
usually  very  slight.    The  electricity  remaining  after  the  first  dis- 
charge is  called  the  residuary  charge.    The  larger  the  jar,  and  the 
more  intense  the  charge,  the  larger  is  this  residuum.    It  is  probably 
explained  as  follows :    The  charge,  at  first  limited  to  the  coatings, 
gradually  diffuses  itself  on  the  uncoated  glass  for  a  little  distance, 
according  to  the  intensity  of  the  charge  and  the  length  of  time 
the  jar  remains  charged.    At  the  first  discharge,  only  the  elec- 
tricity which  is  in  contact  with  the  coating  is  taken  off,  and  that 
which  lies  on  the  uncoated  glass  slowly  diffuses  itself  back  again, 
and  is  conducted  over  the  whole  coated  surface ;  so  that,  after  the 
lapse  of  a  minute  or  two,  a  sensible  discharge  occurs  on  applying 
the  rod  a  second  time. 

427.  The  Electric  Battery. — Leyden  jars  are  made  of  vari- 
ous sizes,  from  a  half-pint  to  one  or  two  gallons.    But  when  a 
great  amount  of  surface  is  needed,  it  is  more  convenient,  and,  in 
case  of  fracture  by  violent  discharge,  more  economical,  to  connect 


262 


STATICAL    ELECTRICITY. 


FIG.  235. 


several  jars,  so  that  they  may  be  used  as  one.  Four,  nine,  twelve, 
or  even  a  greater  number  of  jars,  are 
set  in  a  box  (Fig.  235),  whose  inte- 
rior is  lined  with  tinfoil,  so  as  to  con- 
nect all  the  buter  coatings  together. 
Their  inner  coatings  are  also  con- 
nected, by  wires  joining  all  the 
knobs,  or  by  a  chain  passing  round 
all  the  stems.  Care  is  necessary  in 
discharging  batteries,  that  the  cir- 
cuit is  not  too  short  and  too  perfect, 

since  the  violence  of  discharge  is  liable  to  perforate  the  jars.  A 
chain,  three  or  four  feet  long  in  the  circuit,  will  generally  prevent 
the  accident. 

428.  Different  Routes  of  Discharge. — If  two  or  more  cir- 
cuits are  opened  at  once  between  the  two  coatings  of  a  charged 
jar  or  battery,  the  discharge  will  take  one  or  another,  or  divide  be- 
tween them,  according  to  circumstances.    If  the  circuits  are  alike 
except  in  length,  the  discharge  will  follow  the  shorter.    If  they 
differ  only  in  conducting  quality,  the  electricities  will  take  the 
lest  conductor.    If  the  circuits  are  interrupted,  and  in  all  respects 
alike,  except  that  the  conductors  of  one  are  pointed  at  the  inter- 
ruptions, and  of  the  others  not  pointed,  the  discharge  will  follow 
the  line  which  has  pointed  conductors.    If  the  circuits  are  very  at- 
tenuated (as  very  fine  wire,  or  threads  of  gold-leaf),  the  charge  is 
liable  to  divide  among  them. 

429.  Discharging  Electrometers. — These  are  instruments 
contrived  for  measuring  the  charge  in  the  act  of  discharging  the 
jar.    Fig.  236  represents  Lane's  discharging  electrometer.    D  is  a 


FIG.  236. 


FIG.  237. 


rod  of  solid  glass,  which  holds  the  metallic  rod  and  balls  B  C. 
This  rod,  being  in  a  horizontal  position  at  the  height  of  the  knob 
A9  can  be  placed  at  any  desired  distance  from  it.  Then,  if  the 
circuit  through  which  the  charge  is  to  be  sent  extends  from  the 


THE    UNIT    JAR.  263 

rod  B  C  to  the  external  coating,  the  interval  of  air  between  4  and 
B  is  all  which  prevents  discharge ;  and  as  soon  as  the  charge  is 
increased,  till  its  tension  is  sufficient  to  leap  that  interval,  the  dis- 
charge will  take  place.  The  greater  that  space  is,  of  course  the 
greater  the  charge  must  be,  before  it  will  pass  across.  If  A  and  B 
are  in  contact,  no  charge  at  all  will  collect. 

The  unit  jar  is  used  to  measure  the  charge  of  another  jar,  by 
conveying  to  it  successive  equal  charges  of  its  own.  A  B  (Fig. 
237)  is  the  instrument,  consisting  of  a  small  open  jar,  placed  hori- 
zontally on  an  insulating  stand,  B.  From  the  metallic  part  of 
the  support,  the  bent  rod  and  ball,  C,  come  near  to  D,  the  rod  of 
the  inner  coating,  and  can  be  turned  so  as  to  increase  or  diminish 
its  distance.  Let  the  knob,  /),  be  near  the  prime  conductor,  and 
that  of  the  outer  coating  near  the  top  of  the  jar,  E,  which  is  to  be 
charged,  the  outside  of  the  latter  being  in  communication  with 
the  earth.  While  A  is  charging,  the  positive  electricity  of  its 
outer  coating  goes  to  the  inner  coating  of  the  large  jar,  and  par- 
tially charges  it  Presently  the  unit  jar  discharges  spontaneously 
across  the  distance  between  D  and  C.  It  is  then  in  the  neutral 
condition,  as  at  first,  and  the  process  is  repeated  till  E  is  charged 
with  the  requisite  number  of  units. 

430.  The  Effect  of  a  Point  Presented  to  an  Electri- 
fied Body. — It  has  been  noticed  (Art.  406)  that  a  pointed  wire 
attached  to  the  prime  conductor  wastes  the  charge  very  quickly, 
because  of  the  accumulation  at  the  point.    The  conductor  loses 
its  charge  just  as  quickly  by  presenting  a  pointed  rod  loivard  it. 
For  the  induced  electricity  of  the  rod  and  person  holding  it  is  in 
like  manner  accumulated  at   the  point,  and  readily  escapes  to 
mingle  with  and  neutralize  its  opposite  in  the  prime  conductor. 
Thus  the  charge  disappears  at  once.    In  a  similar  way  is  to  be  ex- 
plained the  use  of  the  points  on  the  prime  conductor  presented  to 
the  glass  plate.    When  the  two  electricities  are  separated  at  the 
surface  of  contact  between  the  plate  and  rubbers,  the  plate  is  posi- 
tively electrified.    This  positive  charge  acts  inductively  on  the 
prime  conductor,  attracting  the  negative  kind  to  the  points,  where 
it  passes  off  and  neutralizes  what  is  on  the  plate,  and  leaves  a  pos- 
itive charge  on  the  prime  conductor. 

431.  The  Gold-Leaf  Electroscope. — The  principle  of  in- 
duction explains  the  construction  of  some  other  instruments  be- 
sides the  Leyden  jar ;  as  the  gold-leaf  electroscope,  the  electrical 
condensers,  and  the  electrophorus.    The  first  has  been  already 
described  (Art.  387) ;  we  have  only  to  explain  its  operation  by  the 
principle   of  induction.      Let  a    body  positively  electrified    be 
brought  within  a  few  feet  of  the  knob.    It  attracts  the  negative 


264 


STATICAL    ELECTRICITY. 


from  the  leaves  into  the  knob,  and  repels  the  positive  from  the 
knob  into  the  leaves;  they  are  thus  electrified  alike,  and  repel 
each  other.  If  the  charged  body  is  brought  so  near  that  the  leaves 
touch  the  conductors,  which  are' placed  on  the  sides  of  the  cylin- 
der, and  discharge  their  induced  electricity  to  them,  then  they 
collapse.  After  this,  they  will  diverge  again,  whether  the  electri- 
fied body  is  brought  still  nearer,  or  withdrawn ;  if  brought  nearer, 
they  diverge  by  means  of  a  new  portion  of  positive,  repelled  from 
the  knob ;  if  withdrawn,  they  diverge  by  the  return  of  negative 
electricity  from  the  knob,  which  is  no  longer  neutralized  by  the 
positive,  since  the  latter  has  been  discharged  to  the  earth. 

432.  The  Electrical  Condenser. — Instruments  called  by 
this  name  are  intended  for  the  accumulation  of  electricity  from 
some  feeble  source,  until  it  may  be  rendered  sensible.     The  most 
delicate  is  the  gold-leaf  condenser.    Suppose  the  gold-leaf  electro- 
scope to  have  a  disk,  J,  instead  of  a  knob  on  the  top  (Fig.  238). 
Another  disk,  B,  is  furnished  with  an  insulating  handle, 

and  between  the  disks  is  placed  the  thinnest  possible 
non-conductor,  as  a  film  of  varnish.  Bring  the  finger 
in  contact  with  the  under-side  of  A,  to  connect  it  with 
the  earth.  Then  bring  to  the  upper  side  of  B  the  source 
of  feeble  electricity  (as,  for  example,  a  piece  of  copper, 
after  being  touched  to  a  piece  of  zinc),  the  small  quan- 
tity of  electricity  imparted  to  B  induces  an  equal 
amount  of  the  opposite  in  A9  drawn  in  from  the  earth. 
After  the  disks  have  touched  each  other  again,  a  second 
contact  upon  B  repeats  the  action ;  and  when  this  has 
been  done  a  great  number  of  times,  there  are  condensed  on  the  two 
sides  of  the  varnish  small  charges  which  are  held  in  that  state  by 
induction.  As  yet,  the  gold-leaves  are  at  rest ;  but  on  removing 
the  finger  from  A,  and  taking  up  B  by  the  insulating  handle,  the 
electricity  condensed  in  A  is  set  free,  flows  down  to  the  leaves  and 
repels  them,  thus  rendering  the  accumulation  perceptible. 

433.  The  Electrophorus. — This  is  a  very  simple  electrical 
machine  for  giving  the  spark.    It  consists  FlG 

of  a  circular  cake  of  resin  in  a  wooden 
base,  A  (Fig.  239),  and  a  metallic  disk,  B, 
having  a  glass  handle.  Excite  the  resin  by 
fur  or  flannel ;  set  the  disk  B  upon  it,  and 
touch  the  latter  with  the  finger.  The  disk 
now  has  a  disguised  charge  of  positive  elec- 
tricity, drawn  in  by  the  negative  charge  of 
the  plate  through  the  finger.  On  lifting 
the  disk  by  the  handle,  its  charge  is  set  free,  and  may  be  taken 


THE    INDUCTION    MACHINE. 


265 


off  in  a  brilliant  spark.  Set  the  disk  down  again,  touch  it,  and 
lift  it,  and  the  same  thing  occurs,  even  hundreds  or  thousands  of 
times,  and,  after  standing  for  hours,  is  ready  to  operate  still  in 
the  same  way. 

This  case  of  inductive  action  seems  at  first  perplexing,  because 
there  is  no  glass  plate,  no  film  of  varnish,  no  non-conductor  of 
any  kind,  between  the  two  opposite  electricities  of  the  resin  and 
disk.  Why  then  do  they  not  at  once  mingle,  and  neutralize  each 
other  ?  It  is  simply  because  the  resin  is  an  excited  body,  the  neg- 
ative electricity  having  been  developed  upon  it  by  friction.  When 
we  touch  the  finger  to  the  disk,  the  positive  that  enters  does  meet 
the  negative,  and  neutralize  it  for  the  time  being ;  but  on  separa- 
ting the  plate  and  disk,  the  electricities  also  separate,  as  is  always 
the  case  when  an  electric  and  the  rubber  are  removed  from  each 
other  after  friction.  Thus  one  charge  on  the  resin  may  be  made 
to  induce  any  number  of  successive  charges  upon  the  disk. 

434.  The  Induction  Machine.— This  instrument  (Fig.  240), 
known  also  as  the  Holtz  machine,  from  the  name  of  the  inventor, 

FIG.  240. 


develops  electricity  with  great  rapidity  without  friction,  except  as 
it  is  employed  for  a  moment  at  first  to  electrify  one  of  the  sectors,  by 
the  side  of  which  the  plate  revolves.  This  electrified  sector  acts  in- 
ductively on  the  successive  portions  of  the  revolving  plate,  with- 
out losing  sensibly  its  own  electric  charge,  just  as  the  electeopho- 


266  STATICAL    ELECTRICITY. 

rus  plate  induces  charge  after  charge  on  the  metallic  disk,  without 
loss  to  itself.* 

435.  The  Leichtenberg  Figures.— When  a  spark  of  elec- 
tricity is  laid  upon  a  non-conductor,  it  will,  by  its  own  self-repel- 
lency,  extend  itself  a  little  distance  along  the  surface.  The  Leich- 
tenberg figures  furnish  a  visible  illustration  of  this  fact,  and  also 
show  that  the  two  fluids  diffuse  themselves  in  very  different  forms. 
Lay  down  sparks  of  positive  electricity  from  the  knob  of  the  Ley- 
den  jar  upon  a  plate  of  resin,  and  near  them  some  sparks  of  nega- 
tive electricity.  Then  blow  upon  the  plate  the  mingled  powders 
of  sulphur  and  red-lead.  The  sulphur,  by  the  agitation  of  passing 
through  the  air,  will  be  electrified  negatively,  and  attracted  there- 
fore by  the  positive  sparks ;  the  red-lead,  positively  electrified,  will 
be  attracted  by  the  negative.  Thus  the  spots  on  which  the  elec- 
tricities are  placed  will  appear  in  their  exact  forms  by  means  of 
the  colored  powders  attached  to  them.  The  positive  resemble 
stars,  or  rather  a  group  of  crystals  shooting  out  from  a  nucleus ; 
the  negative  spots  are  circles  with  smooth  edges ;  and  the  size  of 
the  electrified  spots  in  each  case  depends  on  the  quantity  of  elec- 
tricity in  the  spark. 


CHAPTER   IV. 

EFFECTS  OF  ELECTRICAL  DISCHARGES. 

436.  Variety  of  Effects.— Some  of  the  effects  of  electrical 
discharges  have  been  incidentally  noticed  in  the  foregoing  chap- 
ters.   The  bright  light,  the  sharp  sound,  and  the  great  suddenness 
of  the  transmission,  are  remarkable  phenomena  in  every  discharge 
of  a  Leyden  jar  or  battery.    The  various  effects  may  be  classified 
as  luminous,  mechanical,  chemical,  and  physiological. 

437.  Luminous  Effects. — Light  is  seen  only  when  elec- 
tricity is  discharged  in  considerable  quantities  through  an  ob- 
structing medium.    Hence,  no  light  is  perceived  when  it  flows 
through  a  good  conductor,  unless  of  very  small  diameter.     But  if 
there  is  the  least  interruption,  or  if  the  conductor  is  reduced  to  a 
very  slender  form,  then  light  appears  at  the  interruption,  and  at 
those  parts  which  are  too  small  to  convey  the  electricity.    Thus, 

*  The  Holtz  machine  has  been  greatly  improved  by  Mr.  E.  S.  Ritchie,  of 
Boston.  The  figure  presents  this  improved  form.  Physicists  are  not  fully 
agreed  ras  to  the  mode  of  explaining  all  the  phenomena  of  this  machine. 


LUMINOUS    FIGURES.  267 

the  discharge  of  a  battery  through  a  chain  gives  a  brilliant  scintil- 
lation at  every  point  of  contact  between  the  links. 

438.  Modifications  of  the  Light — The  length,  color,  and 
form  of  the  electric  spark  vary  with  the  nature  and  form  of  the 
conductors  between  which  it  passes,  and  with  the  quality  of  the 
medium  interposed  between  them.        • 

Electrical  sparks  are  more  brilliant  in  proportion  as  the  sub- 
stances between  which  they  occur  are  better  conductors.  A  spark 
received  from  the  prime  conductor  upon  a  large  metallic  ball  is 
short,  straight,  and  white;  on  a  small  ball  it  is  longer,  and 
crooked ;  received  on  the  knuckle,  a  less  perfect  conductor,  the 
middle  part  is  purplish ;  on  wood,  ice,  a  wet  plant,  or  water,  it  is 
red. 

From  a  point  positively  electrified,  the  electricity  passes  in  the 
form  of  a  faint  brush  or  pencil  of  rays ;  a  point  connected  with 
the  negative  side  exhibits  a  luminous  star. 

When  electricity  passes  through  rarefied  air,  the  light  becomes 
faint,  and  is  generally  changed  in  color.  The  electrical  spark,  which 
in  common  air  is  interrupted,  narrow,  and  white,  becomes,  as  the 
rarefaction  proceeds,  continuous,  diffused,  and  of  a  violet  color, 
which  tint  it  retains  as  long  as  it  can  be  seen.  If  a  battery  is  dis* 
charged  through  a  tube  several  feet  long,  nearly  exhausted  of  air, 
the  whole  space  is  filled  with  a  rich  purple  light.  The  sparks 
from  the  machine,  conveyed  through  the  same  tube,  exhibit  flash- 
ings and  tints  exceedingly  resembling  the  Aurora  Borealis. 

The  Geissler  tubes  are  tubes  of  complex  forms,  and  containing 
a  slight  trace  of  some  gas  or  vapor,  which  show  various  colors 
and  intensities  of  electric  light,  according  to  the  kind  of  gas,  the 
diameter  of  the  parts,  and  the  quality  of  the  glass.  The  electricity 
is  conveyed  into  the  tubes  by  platinum  wires  sealed  into  their  ex- 
tremities. 

Various  colors  are  obtained  by  sending  charges  through  differ- 
ent substances.  An  egg  is  bright  crimson ;  the  pith  of  cornstalk, 
orange ;  fluor-spar,  green ;  and  loaf-sugar,  white  and  phosphor- 
escent. 

439.  Luminous  Figures. — Metallic  conductors,  if  of  suffi- 
cient size,  transmit  electricity  without  any  luminous  appearance, 
provided  they  are  perfectly  continuous ;  but  if  they  are  separated 
in  the  slightest  degree,  a  spark  will  occur  at  every  separation. 
On  this  principle,  various  devices  are  formed,  by  pasting  a  narrow 
band  of  tinfoil  on  glass,  in  the  required  form,  and  cutting  it  across 
with  a  penknife,  where  we  wish  sparks  to  appear.    If  an  inter- 
rupted conductor  of  this  kind  be  pasted  round  a  glass  tube  in  a 
spiral  direction,  and  one  end  of  the  tube  be  held  in  the  hand,  and 


2G8 


STATICAL    ELECTRICITY. 


FIG.  241. 


the  other  be  presented  to  an  electrified  conductor,  a  coil  of  bril- 
liant points  surrounds  the  tube.  Words,  flowers,  and  other  com- 
plicated forms,  are  also  produced  nearly  in  the  same  manner,  by  a 
suitable  arrangement  of  interruptions  in  a  narrow  line  of  tinfoil, 
running  back  and  forth  on  a  plate  of  glass. 

4'1Q.  Mechanical  Effects.—  Powerful  electric  discharges 
through  imperfect  conductors  produce  certain  mechanical  effects, 
such  as  perforating,  tearing,  or  breaking  in  pieces,  which  are  all 
due  to  the  sudden  and  violent  repulsion  between  the  electrified 
particles. 

A  discharge  through  the  air  is  supposed  to  perforate  it.  If 
the  air  through  which  the  spark  is  passed  lies  partially  inclosed 
between  two  bodies  which  are  easily  moved,  the  force  by  which 
the  air  is  rent  will  drive  them  asunder.  Thus,  a  little  block  may 
be  driven  out  from  the  foundation  of  a 
miniature  building,  and  the  whole  be  top- 
pled down.  But  this  enlargement  of  in- 
closed air  is  best  seen  in  Kinnersley's  air 
thermometer  (Fig.  241).  As  the  spark 
passes  between  the  knobs  in  the  large 
lube,  the  air  confined  in  it  is  suddenly 
driven  asunder,  so  as  to  press  the  water 
which  occupies  the  lower  part  two  or 
three  inches  up  the  tube,  as  represented. 
As  soon  as  the  discharge  has  occurred, 
the  water  quietly  returns  to  its  level.  The 
sharp  sound  which  is  produced  by  the  dis- 
charge of  a  Leyden  jar  is  due  to  the  sud- 
den compression  of  the  air,  and  also  to  the 
collapse  which  immediately  succeeds. 

The  path  of  the  electric  spark  through 
the  air,  when  short,  is  straight;  but  if 
more  than  about  four  inches  long,  is  usually  crinkled.  This  is 
supposed  to  arise  from  the  condensation  of  the  air  before  it,  by 
which  it  is  continually  turned  aside. 

When  the  charge  is  passed  through  a  thick  card,  or  the  cover 
of  a  book,  a  hole  is  torn  through  it,  which  presents  the  rough  ap- 
pearance of  a  bur  on  each  side.  By  means  of  the  battery,  a  quire  of 
strong  paper  may  be  perforated  in  the  same  manner;  and  such  is 
the  velocity  with  which  the  fluid  moves,  that  if  the  paper  be  freely 
suspended,  not  the  least  motion  is  communicated  to  it.  Pieces  of 
hard  wood,  of  loaf-sugar,  and  brittle  mineral  substances,  are  split 
in  two,  or  shivered  to  pieces,  by  an  intense  charge  of  a  battery. 
But  good  conductors  of  much  breadth  are  not  thus  affected.  The 


CHEMICAL    EFFECTS.  269 

charge,  as  it  is  transmitted,  passes  over  the  whole  body,  instead  of 
being  concentrated  in  any  one  line.  But  if  liquids  which  are  good 
conductors  are  closely  confined  on  every  side,  they  show  that  a 
violent  expansion  is  produced  by  a  discharge.  Thus,  when  a 
charge  is  sent  through  water  confined  in  a  small  glass  tube  or 
ball,  the  glass  is  shattered  to  pieces ;  and  mercury  in  a  thick  cap- 
illary tube  is  expanded  with  a  force  sufficient  to  splinter  the  glass. 

441.  Chemical  Effects. — These  are  various :  combustion  of 
inflammable  bodies;  oxydation,  fusion,  and  combustion  of  metals; 
separation  of  compounds  into  their  elements ;  reunion  of  elements 
into  compounds. 

Ether  and  alcohol  may  be  inflamed  by  passing  the  electric 
spark  through  them ;  phosphorus,  resin,  and  other  solid  combus- 
tible bodies,  may  be  set  on  fire  by  the  same  means ;  gunpowder 
and  the  fulminating  powders  may  be  exploded,  and  a  candle  may 
be  lighted.  Gold-leaf  and  fine  iron  wire  may  be  burned,  by  a 
charge  from  the  battery.  Wires  of  lead,  tin,  zinc,  copper,  plati- 
num, silver,  -and  gold,  when  subjected  to  the  charge  of  a  very  large 
battery,  are  burned,  and  converted  into  oxides. 

The  same  agent  is  also  capable  of  restoring  these  oxides  to 
their  simple  forms.  Water  is  decomposed  into  its  gaseous  ele- 
ments, and  these  elements  may  again  be  reunited  to  form  water. 
By  passing  a  great  number  of  electric  charges  through  a  confined 
portion  of  air,  the  oxygen  and  nitrogen  are  converted  into  nitric 
acid.  The  ozone  which  is  almost  always  perceived  in  connection 
with  electrical  experiments  is  to  be  considered  as  one  of  the  chem- 
ical effects  of  electricity. 

Galvanic  electricity  is  a  form  of  this  agent  much  better  adapted 
than  frictional  electricity  to  produce  chemical  as  well  as  magnetic 
effects. 

442.  Physiological  Effects. — The  shock  experienced  by  the 
animal  system,  when  the  charge  of  a  jar  passes  through  it,  has 
been  already  mentioned. 

A  slight  charge  of  the  Leyden  jar,  passed  through  the  body 
from  one  hand  to  the  other,  affects  only  the  fingers  or  the  wrists ; 
a  stronger  charge  convulses  the  large  muscles  of  the  arms ;  a  still 
greater  charge  is  felt  in  the  breast,  and  becomes  somewhat  pain- 
ful. The  charge  of  a  large  battery  is  sufficient  to  destroy  life,  if  it 
be  sent  through  the  vital  organs.  By  connecting  the  chains 
which  are  attached  to  the  jar  with  insulating  handles,  it  is  easy  to 
pass  shocks  through  any  particular  joint,  muscle,  or  other  part  of 
the  body,  as  is  frequently  done  for  medical  purposes. 

The  charge  may  be  passed  through  a  great  number  of  persons 
at  the  same  time.  Hundreds  of  individuals,  by  joining  hands, 


270  STATICAL    ELECTRICITY. 

have  received  the  shock  at  once,  though  there  is  more  difficulty  in 
passing  a  charge  of  given  intensity  as  the  number  is  increased. 

If  the  spark  is  taken  by  a  person  from  the  prime  conductor, 
the  quantity  is  not  sufficient,  unless  the  conductor  is  of  extraordi- 
nary size,  to  produce  what  is  called  the  shock ;  a  pricking  sensa- 
tion in  the  flesh  where  the  spark  strikes,  and  a  slight  spasm  of  the 
muscle,  is  all  that  is  noticeable.  A  person  may  make  his  own 
body  a  part  of  the  prime  conductor  by  standing  on  an  insulating 
stool — that  is,  a  stool  having  glass  legs,  and  touching  the  conduc- 
tor of  the  machine.  This  occasions  no  sensation  at  all,  except 
wltat  arises  from  the  movement  of  the  hair,  in  yielding  to  the  re- 
pellency  of  the  fluid.  If  another  person  takes  the  spark  from  him, 
the  prick  is  more  pungent,  as  the  quantity  is  larger  than  in  the 
prime  conductor  alone. 

443.  Velocity  of  Electricity. — This  is  so  great  that  no  ap- 
preciable time  is  occupied  in  any  case  of  discharge.  When  we 
seem  to  see  lightning  move/rom  the  cloud  to  the  earth,  we  find 
that  such  a  progress  is  imagined,  not  perceived ;  for,  by  a  little 
effort,  we  can  just  as  well  learn  to  see  it  pass  from  the  earth  to  the 
cloud. 

Wheatstone  a  few  years  since  devised  an  ingenious  method  of 
measuring  the  time  in  which  electricity  passes  over  a  wire  only 
half  a  mile  long.  The  wire  was  so  arranged  that  three  interrup- 
tions, one  near  each  end,  and  one  in  the  middle  of  the  wire,  were 
brought  side  by  side.  When  the  discharge  of  a  jar  was  transmitted, 
the  sparks  at  these  interruptions  were  seen  by  reflection  in  a 
swiftly  revolving  mirror.  An  exceedingly  small  difference  of 
time  between  the  passage  of  those  interruptions  could  be  easily 
perceived  by  the  displacement  of  the  sparks  as  seen  in  the  whirling 
mirror.  The  amount  of  observed  displacement  and  the  known 
rate  of  revolution  of  the  mirror,  would  furnish  the  interval  of  time 
occupied  by  the  electricity  in  passing  from  one  interruption  to  the 
next.  By  a  series  of  experiments,  Wheatstone  arrived  at  the  con- 
clusion that,  on  copper  wire,  one-fifteenth  of  an  inch  in  diameter, 
electricity  moves  at  the  rate  of  288,000  miles  per  second,  a  velocity 
much  greater  than  that  of  light. 

Galvanic  electricity  moves  very  much  slower.  Its  rate  on  iron 
wire,  of  the  size  usually  employed  for  telegraph  lines,  is  about 
16,000  miles  per  second. 


ELECTRICITY    IN    THE    AIR.  271 

CHAPTER   V. 

ATMOSPHERIC  ELECTRICITY.— THUNDER   STORMS. 

444.  Electricity  in   the  Air. — The  atmosphere  is  always 
more  or  less  electrified,  sometimes  pos'tively,  sometimes  negatively. 
This  fact  is  ascertained  by  several  different  forms  of  apparatus. 
For  the  lower  strata,  it  is  sufficient  to  elevate  a  metallic  rod  a  few 
feet  in  length,  pointed  at  the  top,  and  insulated  at  the  bottom. 
With  the  lower  extremity  is  connected  an  electroscope,  which  in- 
dicates the  presence  and  intensity  of  the  electricity.    For  experi- 
ments on  the  electricity  of  higher  portions,  a  kite  is  employed, 
with  the  string  of  which  is  intertwined  a  fine  metallic  wire.    The 
lower  end  of  the  string  is  insulated  by  fastening  it  to  a  support  of 
glass,  or  by  a  cord  of  silk.    If  a  cloud  is  near  the  kite,  the  quan- 
tity of  electricity  conveyed  by  the  string  may  be  greatly  increased, 
and  even  become  dangerous.     Cavallo  received  a  large  number  of 
severe  shocks  in  handling  the  kite-string ;  and  Rich  man,  of  Peters- 
burgh,  was  killed  by  a  discharge  of  electricity  which  came  down 
the  rod  which  he  had  arranged  for  his  experiments,  but  which  was 
not  provided  with  a  conductor  near  by  it,  for  taking  off  extra 
charges. 

The  electricity  of  the  atmosphere  is  most  developed  when  hot 
dry  weather  succeeds  a  series  of  rainy  days,  or  the  reverse ;  and 
during  a  single  day,  the  air  is  most  electrical  when  dew  is  begin- 
ning to  form  before  sunset,  or  when  it  begins  to  exhale  after  sun- 
rise. In  clear,  steady  weather,  the  electricity  is  generally  positive ; 
biit  in  falling  or  stormy  weather,  it  is  frequently  changing  from 
positive  to  negative,  and  from  negative  to  positive. 

445.  Thunder-Storms. — Thunder-clouds  are,  of  all  atmos- 
pheric bodies,  the  most  highly  charged  with  electricity ;  but  all 
single,  detached,  or  insulated  clouds,  are  electrified  in   greater 
or  less  degrees,  sometimes  positively  and  sometimes  negatively. 
When,  however,  the  sky  is  completely  overcast  with  a  uniform 
stratum  of  clouds,  the  electricity  is  much  feebler  than  in  the  single 
detached  masses  before  mentioned.      And,  since  fogs  are  only 
clouds  near  the  surface  of  the  earth,  they  are  subject  to  the  same 
conditions :  a  driving  fog,  of  limited  extent,  is  often  highly  elec- 
trified. 

Thunder-storms  occur  chiefly  in  the  hottest  season  of  the  year, 
and  after  mid-day,  and  are  more  frequent  and  violent  in  warm 
than  in  cold  countries.  They  never  occur  beyond  75°  of  latitude — 


272  STATICAL    ELECTRICITY. 

seldom  beyond  65°.  In  the  New  England  States  they  usually 
come  from  the  west,  or  some  westerly  quarter. 

The  storm  itself,  including  everything  except  the  electrical  ap- 
pearances, is  supposed  to  be  produced  in  the  same  manner  as 
other  storms  of  wind  and  rain ;  and  the  electricity  is  developed  by 
the  rapid  condensation  of  watery  vapor,  and  by  friction.  Elec- 
tricity is  not  to  be  regarded  as  the  cause,  but  as  a  consequence  or 
concomitant  of  the  storm.  But  the  precipitation  of  vapor  must  be 
sudden  and  copious,  since  when  the  process  is  slow,  too  much  of 
the  electricity  evolved  would  escape  to  allow  of  the  requisite  accu- 
mulation. Also,  if  a  storm-cloud  is  of  great  extent,  it  is  not  likely 
to  be  highly  electrified,  because  the  opposite  electricities,  which 
may  be  developed  in  different  parts  of  it,  have  opportunity  to 
mingle  and  neutralize ;  and  points  of  communication  with  the 
earth  will  here  and  there  occur.  Clouds  of  rapid  formation,  violent 
motion,  and  limited  extent,  are  therefore  most  likely  to  be  thun- 
der-clouds. 

446.  Lightning. — When  a  cloud  is  highly  charged,  it  operates 
inductively  on  other  bodies  near  it,  such  as  other  clouds,  or  the 
earth.    Hence,  discharges  will  occur  between  them.    Lightning 
passes  frequently  between  two  clouds,  or  even  between  two  parts 
of  the  same  cloud,  in  which  opposite  electricities  are  so  rapidly  de- 
veloped that  they  cannot  mingle  by  conduction.     But,  in  general, 
the  discharges  of  lightning  take  place  between  the  electrified  cloud 
and  the  earth,  whose  nearer  part  is  thrown  into  the  opposite  elec- 
trical state  by  induction.    It  is  supposed  that,  in  some  instances, 
a  discharge  occurs  between  two  distant  clouds  by  means  of  the 
earth,  which  constitutes  an  interrupted  circuit  between  them. 
The  crinkled  form  of  the  path  of  lightning  is  explained  in  the 
same  way  as  that  of  the  spark  from  the  machine,  and  the  thunder 
is  caused  by  the  simultaneous  rupture  and  collapse  of  air  in  all 
parts  of  the  line  of  discharge.    The  words  chain-lightning,  sheet- 
lightning,  and  heat-lightning,  are  supposed  not  to  indicate  any  real 
differences  in  the  lightning  itself,  but  only  in  the  circumstances 
of  the  person  who  observes  it.    If  the  crinkled  line  of  discharge  is 
seen,  it  is  chain  or  fork  lightning;  if  only  the  light  which  pro- 
ceeds from  it  is  noticed,  it  is  s/^-lightning;  if,  in  the  evening, 
the  thunder-storm  is  so  far  distant  that  the  cloud  cannot  be  seen, 
nor  the  thunder  heard,  but  only  the  light  of  its  discharges  can  be 
discerned  in  the  horizon,  it  is  frequently  called  heat-ligh tiling. 

447.  Identity  of  Lightning  and  Electrical  Discharges.— 

Franklin  was  the  first  to  point  out  the  resemblances  between  the 
phenomena  of  lightning  and  those  of  frictional  electricity.  He 
was  also  the  first  to  propose  the  performance  of  electrical  experi- 


LIGHTNING-RODS.  273 

ments  by  means  of  electricity  drawn  from  the  clouds.  The  points 
of  resemblance  named  by  Franklin  were  these :  1.  The  crinkled 
form  of  the  path.  2.  Both  take  the  most  prominent  points. 
3.  Both  follow  the  same  materials  as  conductors.  4.  Both  inflame 
combustible  substances.  5.  They  melt  metals  in  attenuated  forms. 
6.  They  fracture  brittle  bodies.  7.  Both  have  produced  blindness. 
8.  Both  destroy  animal  life.  9.  Both  affect  the  magnetic  needle  in 
the  same  manner.  In  1752,  he  obtained  electricity  from  a  thunder- 
cloud by  a  kite,  and  charged  jars  with  it,  and  performed  the  usual 
electrical  experiments. 

448.  Lightning-Rods. — Franklin  had  no  sooner  satisfied  him- 
self of  the  identity  of  electricity  and  lightning  than,  with  his  usual 
sagacity,  he  conceived  the  idea  of  applying  the  knowledge  acquired 
of  the  properties  of  the  electric  fluid  so  as  to  provide  against  the 
dangers  of  thunder-storms.  The  conducting  power  of  metals,  and 
the  influence  of  pointed  bodies,  to  transmit  the  fluid,  naturally 
suggested  the  structure  of  the  lightning-rod.  The  experiment 
was  tried,  and  has  proved  completely  successful;  and  probably  no 
single  application  of  scientific  knowledge  ever  secured  more  celeb- 
rity to  its  author. 

Lightning-rods  are  often  constructed  of  wrought  iron,  about 
three-fourths  of  an  inch  in  diameter.  The  parts  may  be  made 
separate,  but,  when  the  rod  is  in  its  place,  they  should  be  joined 
.  together  so  as  to  fit  closely,  and  to  make  a  continuous  surface, 
since  the  fluid  experiences  much  resistance  in  passing  through 
links  and  other  interrupted  joints.  At  the  bottom  the  rod  should 
terminate  in  two  or  three  branches,  going  off  in  a  direction  from 
the  building,  and  descending  to  such  a  depth  that  they  will  reach 
permanent  moisture.  At  top  the  rod  should  be  several  feet  higher 
than  the  highest  parts  of  the  building.  It  is  best,  when  practica- 
ble, to  attach  it  to  the  chimney,  which  needs  peculiar  protection, 
both  on  account  of  its  prominence  and  because  the  products  of  the 
combustion,  smoke,  watery  vapor,  &c.,  are  conductors  of  elec- 
tricity. For  a  similar  reason,  a  kitchen  chimney,  being  that  in 
which  the  fire  is  kept  during  the  season  of  thunder-storms,  re- 
quires to  be  especially  protected.  The  rod  is  terminated  above  in 
one  or  more  sharp  points ;  and  as  these  points  are  liable  to  lose 
their  sharpness,  and  have  their  conducting  power  impaired  by 
rust,  they  are  protected  from  corrosion  by  being  covered  with 
gold-leaf  or  silver-plate.  Rods  may  be  made  of  smaller  size  than 
above  described;  but  if  so,  there  should  be  a  proportionally 
greater  number.  It  is  well  to  connect  with  the  rods,  and  with  the 
earth,  all  extended  conductors  upon  or  within  the  building,  such 
as  metallic  coverings  of  roofs,  water  conductors,  bundles  of  bell- 
18 


274  STATICAL    ELECTRICITY. 

wires,  &c. ;  in  order  that  large  discharges  may  have  opportunity 
to  divide,  and  take  several  circuits,  without  doing  injury  at  the 
non-conducting  intervals. 

449.  In  what  way  Lightning -Rods  Afford  Protection.— 

Lightning-rods  are  of  service,  not  so  much  in  receiving  a  discharge 
when  it  comes,  as  in  diminishing  the  number  of  discharges  in  their 
vicinity.  They  continually  carry  on  a  silent  communication  be- 
tween the  two  electricities,  which  are  attracting  each  other,  one 
in  the  cloud,  the  other  in  fche  earth ;  so  that  a  village  well  fur- 
nished with  rods  has  few  discharges  of  lightning  in  it.  All  tall 
pointed  objects,  like  spires  of  churches  and  masts  of  ships,  exert  a 
similar  influence,  though  in  a  less  degree,  because  not  so  good 
conductors. 

During  a  thunder-storm,  or  immediately  after  it,  if  a  person 
can  be  near  the  top  of  a  high  rod,  he  will  sometimes  hear  the  hiss- 
ing sound  of  electricity  escaping  from  it,  as  from  a  point  attached 
to  the  prime  conductor  of  a  machine.  In  the  same  circumstances, 
if  it  were  quite  dark,  he  would  probably  see  the  brush  or  star  of 
light  on  the  point.  The  statement  of  Caesar  in  his  Commentaries, 
"that  the  points  of  the  soldiers'  darts  shone  with  light  in  the 
night  of  a  severe  storm,"  probably  refers  to  the  visible  escape  of 
electricity  from  the  weapons  as  from  lightning-rods. 

450.  Protection  of  the  Person. — Silk  dresses  are  some- 
times worn  with  the  view  of  protection,  by  means  of  the  insula-  . 
tion  they  afford.     They  cannot,  however,  be  deemed  effectual  un- 
less they  completely  envelop  the  person ;  for  if  the  head  and  the 
extremities  of  the  limbs  are  exposed,  they  will  furnish  so  many 
avenues  as  to  render  the  insulation  of  the  other  parts  of  the  sys- 
tem of  little  avail.    The  same  remark  applies  to  the  supposed  se- 
curity that  is  obtained  by  sleeping  on  a  feather  bed.    Were  the 
person  situated  within  the  bed,  so  as  to  be  entirety  enveloped  by 
the  feathers,  they  would  afford  some  protection ;  but  if  the  person 
be  extended  on  the  surface  of  the  bed,  in  the  usual  posture,  with 
the  head  and  feet  nearly  in  contact  with  the  bedstead,  he  would 
rather  lose  than  gain  by  the  non-conducting  properties  of  the  bed, 
since,  being  a  better  conductor  than  the  bed,  the  charge  would 
pass  through  him  in  preference  to  that.     If  the  bedstead  were  of 
iron,  its  conducting  quality  would  probably  be  a  better  protection 
than  the  insulating  property  of  the  feathers,  since,  by  taking  the 
charge  itself,  it  would  keep  it  away  from  the  person.     So,  a  man's 
garments  soaked  with  rain  have  been  known  to  save  his  life,  being 
a  better  conductor  than  his  body.    Animals  under  trees  are  pecu- 
liarly exposed,  because  the  trees  by  their  prominence  are  liable  to 
be  the  channels  of  communication  for  the  electric  discharge,  and 


DAMAGE    BY    LIGHTNING.  275 

the  animal  body,  so  far  as  ifc  reaches,  is  a  better  conductor  than 
the  tree.  Tall  trees,  however,  situated  near  a  dwelling-house,  fur- 
nish a  partial  protection  to  the  building,  being  both  better  con- 
ductors than  the  materials  of  the  house,  and  having  the  advantage 
of  superior  elevation. 

451.  How  Lightning  Causes  Damage. — The  word  strike, 
which  is  used  with  reference  to  lightning,  conveys  no  correct  idea 
of  the  nature  of  the  movement  of  electricity,  or  of  the  injury 
which  it  causes.  One  kind  of  electricity,  developed  in  a  cloud, 
causes  the  other  to  be  accumulated  by  induction  in  the  part  of  the 
earth  nearest  to  it.  These  electricities  strongly  attract  each  other ; 
consequently,  that  in  the  earth  presses  upward  into  all  prominent 
conducting  bodies  toward  the  other ;  and,  if  those  bodies  are  nu- 
merous, high,  the  best  of  conductors,  and  terminated  by  points, 
the  electricity  will  flow  off  from  them  abundantly,  and  mingle 
with  its  opposite  in  the  air  above ;  and  thus  discharges  are  in  a 
great  degree  prevented.  But  if  these  channels  for  silent  commu- 
nication are  not  furnished,  the  quantity  of  electricity  will  increase, 
till  the  strength  Of  attraction  becomes  so  great  that  the  fluid  will 
break  its  way  through  the  air,  usually  from  some  prominent  ob- 
ject, as  a  building  or  tree,  and  thus  the  union  of  the  two  elec- 
tricities takes  place.  The  building  or  tree  in  this  case  is  said  to 
ba  struck  ~by  lightning ;  it  is  rent,  or  otherwise  injured,  by  the 
great  quantity  of  electricity  which  passes  violently  through  it,  in 
an  inconceivably  short  space  of  time.  The  effects  produced  are 
exactly  like  those  caused  by  discharges  of  the  electrical  battery, 
on  a  greatly  enlarged  scale.  The  charge  of  a  large  battery,  taken 
through  the  body  in  the  usual  way,  would  prostrate  a  person  by 
the  violence  of  the  shock ;  but  the  same  charge,  if  allowed  to  oc- 
cupy a  few  seconds  in  passing  by  means  of  a  point,  would  not  be 
felt  at  all. 

Fulgurites  are  tubes  of  silicious  matter  formed  in  the  ground, 
where  lightning  has  struck  in  sandy  soil,  and  melted  the  sand 
around  its  path. 


PART   VII. 

ELECTRICITY, 


CHAPTER  I. 

THE  GALVANIC  CURRENT,  AND  APPARATUS  FOR  PRODUCING  IT. 

452.  Electricity  Developed  by  Chemical  Action.— In 

a  glass  vessel  (Fig.  242)  containing  a  mixture  of  one  part  of  sul- 
phuric acid  and  seven  or  eight  parts  of 
water,  put  two  plates,  one  of  copper,  C,  FlG- 

and  the  other  of  zinc,  Z,  to  each  of  which 
is  soldered  a  copper  wire.  On  bringing 
the  extreme  ends  of  the  wires  together,  a 
feeble  flow  of  electricity  will  take  place 
through  the  wires,  the  plates,  and  the 
liquid.  This  is  called  the  galvanic  or  vol- 
taic current  of  electricity.  It  is  developed 
by  the  chemical  action  of  the  acid  on  the 
metals;  and  this  condition  of  electricity 
is  called  galvanic  or  voltaic,  from  Galvani  and  Volta,  two  Italian 
philosophers,  who  made  the  first  discoveries  of  importance  in  this 
branch  of  science.  It  is  also  called  dynamical  electricity,  for  rea- 
sons to  be  mentioned  hereafter. 

453.  Definitions. — An  element  or  cell  is  a  jar  containing  any 
arrangement  of  substances  for  the  purpose  of  obtaining  the  gal- 
vanic electricity.    A  lattery  is  a  number  of  elements  properly  con- 
nected with  each  other. 

The  poles  or  electrodes  of  a  cell  or  battery  are  the  extremities 
of  the  wires  where  the  electricities  appear. 

The  circuit  is  the  path  or  conductor  provided  for  the  flow  of 
the  current — that  is,  the  liquid,  the  plates,  and  the  wires.  The 
circuit  is  said  to  be  closed  when  the  wires  are  joined,  so  that  there 
is  a  flow  of  the  current ;  when  they  are  separated,  the  current 
ceases,  and  the  circuit  is  said  to  be  broken,  or  to  be  open. 


CONSTANT    BATTERIES.  377 

454.  The  Essential  Parts  of  an  Element. — An  element 
must  consist  of  two  unlike  substances  (they  are  generally  two  dif- 
ferent metals),  separated  ly  continuous  moisture. 

Volta's  original  battery,  called  the  dry  pile,  consisting  of  alter- 
nate disks  of  copper,  zinc,  and  paper,  was  no  real  exception,  since 
the  paper  absorbed  sufficient  moisture  from  the  atmosphere. 

455.  The  Cell  of  Two  Fluids. — An  element  of  copper, 
zinc,  and  dilute  acid,  already  described,  soon  loses  its  efficiency. 
Improved  batteries,  by  which  a  constant  flow  of  electricity  may  be 
maintained  for  a  considerable  length  of  time,  are  those  in  which 
two  liquids  are  employed,  and  generally  some  other  substance  than 
copper  for  one  of  the  metals.    The  liquids  must  be  separated  by 
some  porous  substance,  which  shall  prevent  them  from  mingling, 
and  at  the  same  time,  being  saturated  by  the  liquid,  shall  not  in- 
terrupt the  necessary  moist  communication  between  the  metals. 

456.  Constant  Batteries. — Batteries  composed  of  cells  con- 
taining two  liquids  are  called  constant,  because  their  action  con- 
tinues for  so  long  a  time  without  sensible  abatement.     Among  the 
best  of  these  is  Grove's  lattery,  one  element  of  which  is  shown  in 
Fig.  243,  which  represents  a  glass  jar  contain- 
ing a  hollow  cylinder  of  zinc,  which  has  a 

narrow  opening  on  one  side  from  top  to  bot- 
tom, that  the  liquid  in  which  it  is  placed  may 
circulate  freely  within  it.  Within  the  zinc  is 
a  cylindrical  cup  of  porous  earthenware,  and 
within  that  is  suspended  a  lamina  of  plati- 
num. One  of  the  circuit  wires  is  in  metallic 
communication  with  the  zinc,  and  the  other 
with  the  platinum,  by  means  of  the  binding 
screws  at  the  top.  The  earthen  cup  is  now 
filled  with  strong  nitric  acid,  while  the  space  outside  of  it,  in 
which  the  zinc  is  placed,  contains  dilute  sulphuric  acid.  It  is 
necessary  to  amalgamate  the  surface  of  the  zinc  with  mercury,  in 
order  to  prevent  the  action  of  the  acid  when  the  circuit  is  broken. 
Bunsen's  lattery  is  the  same  as  Grove's,  except  that  in  it  a 
cylinder  of  carbon  is  used  instead  of  a  leaf  of  platinum,  on  account 
of  the  expense  of  the  latter.  It  is  very  generally  employed  in 
telegraphy.  Fig.  244  is  a  Bunsen  battery  of  ten  cells. 

457.  Direction  of  the  Current. — Both  positive  and  nega- 
tive electricities  are  furnished  by  a  galvanic  battery.     In  one  of 
copper  and  zinc,  the  former  is  found  at  the  extremity  of  the  wiro 
connected  with  the  copper  plate,  which  extremity  is  therefore 
called  the  positive  electrode.     For  a  corresponding  reason,  the 


378 


DYNAMICAL    ELECTRICITY. 


other  electrode  is  the  negative  one.  On  the  supposition  that  elec- 
tricity is  a  fluid  (a  hypothesis  which  is  now  discarded,  though  the 
convenient  terms  which  it  gave  rise  to,  as  current,  flow,  &c.,  are 


FIG.  244. 


retained),  there  are  manifestly  two  currents,  flowing  in  opposite 
directions.  For  the  sake  of  convenience,  only  the  positive  one  is 
spoken  of  as  the  current.  The  direction  in  which  this  passes 
through  the  wires  is  from  the  copper  to  the  zinc. 

458.  Galvanio  and  Frictional  Electricity  Compared.— 

The  electricities  furnished  by  chemical  action  and  hy  friction  are 
undoubtedly  the  same  in  kind.  But  they  differ  iii  that  the  former 
is  produced  in  greater  quantity,  while  the  latter  is  in  a  state  of 
greater  intensity,  or  tension.  This  will  be  understood  by  referring 
to  heat.  The  quantity  of  heat  in  a  warm  room  is  vastly  greater 
than  that  in  the  flame  of  a  lamp ;  yet  the  former  is  agreeable, 
while  the  latter,  if  touched,  causes  severe  pain  by  its  greater  inten- 
sity. In  a  similar  manner,  a  quantity  of  galvanic  electricity  may 
pass  through  the  body  without  harm,  which,  if  it  possessed  the  in- 
tensity of  frictional  electricity,  would  instantly  destroy  life. 

The  word  tension,  or  intensity,  expresses  the  degree  of  force  ex- 
erted by  electricity  in  overcoming  a  given  obstacle,  as  a  break  in  a 
circuit. 

(1)  From  this  difference  in  quantity  and  intensity  results  a 
very  great  difference  in  continuance  of  action.  This  is  indicated  by 
the  terms  dynamical  and  statical.  Galvanic  electricity,  being  pro- 
duced in  prodigious  quantities  and  with  very  feeble  tension,  may 
flow  in  a  steady,  gentle  stream  for  many  hours,  and  is  hence  called 
dynamical.  While  frictional  electricity,  being  small  in  quantity  and 
intense  in  action,  darts  through  an  opposing  medium  instantane- 
ously, and  with  great  violence.  What  motion  it  has  is  therefore 


CONSTANT    BATTERIES.  279 

merely  incidental  to  its  passage  from  one  state  of  rest  to  another. 
Hence  the  propriety  of  the  term  statical. 

(2)  Again,  owing  to  its  low  tension,  galvanic  electricity  will 
traverse  many  thousands  of  feet  of  wire  rather  than  pass  through 
the  thin  covering  of  silk  with  which  the  wire  is  insulated,  and 
which  would  be  but  a  slight  obstacle  in  the  path  of  frictional 
electricity. 

(3)  Analogous  to  the  latter  is  its  inability  to  pass  from  one 
conductor  to  another  in  its  immediate  vicinity.  In  order  to  es- 
tablish the  flow  of  a  current,  the  electrodes  must  first  be  brought 
into  actual  contact,  or  exceedingly  near  to  each  other.  They  may 
then  be  separated  more  or  less,  according  to  the  intensity  of  the 
battery,  without  interrupting  the  current. 

459.  Actual  Amount — Comparisons  have  been  made  of  the 
actual  quantities  of  electricity  obtained  by  chemical  action  and  by 
friction.     Faraday  has  shown  that  to  decompose   one   grain  of 
water  into  its  constituent  elements,  oxygen  and  hydrogen,  requires 
an  amount  of  frictional  electricity  equal  to  the  charge  of  a  Leyden 
battery  with  a  metallic  surface  of  thirty-two  acres,  equal  to  a  very 
powerful  flash  of  lightning.     But  by  a  galvanic  current,  the  same 
result  is  accomplished  in  three  minutes  and  forty-five  seconds. 
From  this  some  idea  may  be  formed  of  the  vast  quantity  of  elec- 
tricity produced  during  the  steady  flow  for  several  hours  of  a 
Grove  or  Bunseu.  battery. 

460.  Quantity  and  Tension  Regulated. — It  may  be  stated 
in  general  that  quantity  increases  with  the  surface  of  metal,  and 
intensity  with  number  of  elements.     Thus,  from  an  element  which 
presents  two  square  feet  of  surface  of  metal  to  the  action  of  the 
acids,  we  obtain  a  greater  quantity  of  electricity  than  from  one 
whose  metallic  surface  is  one  square  foot,  but  no  increase  of  ten- 
sion.    On  the  other  hand,  from  two  elements,  each  of  one  square 
foot  of  surface,  we  find  greater  tension,  but  no  increase  in  quantity. 

461.  Manner  of  Connecting  the  Elements  of  a  Bat- 
tery.— When  quantity  of  electricity  is  desired,  all  the  plates  of 
the  same  name  in  the  several  elements  should  be  united  by  con- 
necting wires,  as,  for  example,  all  the  zinc  plates  together,  and  all 
the  copper  together.    The  battery  thus  becomes  substantially  a 
single  large  cell,  and  is  called  a  quantity  battery. 

When  tension  is  sought  for,  the  zinc  of  one  cell  should  be 
joined  to  the  copper  of  the  next,  and  so  on  through  the  series.  A 
battery  thus  formed  is  called  an  intensity  battery. 

462.  Effects. — The  presence  of  a  galvanic  current  is  indicated 
by  certain  chemical,  physical,  physiological,  or  magnetic  effects. 


280  DYNAMICAL    ELECTRICITY. 

An  example  of  the  first  is  the  decomposition  of  water,  already 
mentioned. 

A  physical  effect  is  the  production  of  light.  When  the  elec- 
trodes are  brought  together,  and  then  separated,  a  spark  is  pro- 
duced of  varying  intensity  and  duration. 

The  shock  which  is  felt  when  the  electrodes  are  held  in  the 
hands,  and  which  affects  more  or  less  of  the  person,  is  a  physio- 
logical effect. 

The  magnetic  properties  of  a  current  will  be  spoken  of  here- 
after. 

463.  Size  of  Battery  for  Required  Results. — The  phys- 
ical or  physiological  results  obtained  from  a  single  element  of  or- 
dinary size  of  any  kind  are  quite  limited.  No  shock  can  be  ob- 
tained from  the  direct  current  of  a  single  cell.  But  a  smart  one  is 
given  by  fifty  Bunsen  cells.  It  is  felt  only  at  the  instant  of  closing 
or  breaking  the  circuit.  A  shock  from  a  battery  of  several  hundred 
cells  would  affect  the  system  painfully,  if  not  dangerously.  Nine 
hundred  cells  of  copper,  zinc,  and  dilute  acid,  furnish  an  arch  of 
flame  between  the  electrodes  six  inches  in  length.  Brilliant  re- 
sults are  also  obtained  from  twenty  Grove  cells. 

Such  magnetic  and  chemical  results  as  require  a  current  of  low 
intensity  and  small  quantity  may  readily  be  obtained  from  a  single 
cell.  Such  are  electrotyping  or  the  deflection  of  the  magnetic 
needle. 


CHAPTER   II. 

ELECTRO-MAGNETISM. 

» 

464.   Helices. — A  wire  bent  in  a  spiral,  as  in  Fig.  245,  is 
called  a  coil  or  helix.    If  the  wire  is  coiled  in  the  direction  of  the 
thread    of    a    common    or 
right-hand  screw  (Art.  136), 
it   is    called  a   right-hand 
helix;    if  in  the  direction 
of  the  thread  of  a  left-hand 

screw,  it  is  called  a  left-hand  FlG-  246- 

helix.    Without  referring  to 
the   screw,   the   distinction 
between  the  right  and  left 
hand  helix  may  be  described  thus:   When  a  person  looks  at  a 
helix  in  the  direction  of  its  length,  if  the  wire,  as  it  is  traced  from 


THE    SOLENOID. 

him,  winds  from  the  left  over  to  the  right,  it  is  a  right-hand  helix 
(Fig.  245) ;  if  from  the  right  over  to  the  left,  a  left-hand  helix 
(Fig.  246). 

465.  The  Solenoid. — Let  a  helix  be  constructed  as  in  Fig. 
247,  in  which  the  ends  are  turned  back  through  the  coil,  metallic 
contact  being  avoided  through- 
out; tins  is  called  a  solenoid —  FIG.  247 

that  is,  a  tubular  or  channel- 
shaped  magnet.  Next,  let  the 
electrodes  p  and  n  of  a  battery 
be  furnished  with  sockets,  one 
vertically  above  the  other,  in 
which  the  two  ends  of  the  helix 
wire  are  placed.  The  solenoid 
is  then  free  to  turn  nearly  a  whole  revolution  around  a  vertical 
axis,  at  the  same  time  that  a  current  is  passing  through  it.  The 
helix  is  supposed  to  be  a  left-hand  one,  and  is  so  connected  with 
the  battery  that  the  current  passes  through  it  from  N  to  S,  and 
therefore  around  it  from  right  over  to  left. 

While  the  current  flows,  the  following  phenomena  may  be  ob- 
served : 

1.  If  a  magnet  be  brought  near  it,  JV  will  be  attracted  by  the 
south  pole,  and  S  by  the  north  pole.    If,  instead  of  a  magnet, 
another  solenoid  be  presented  to  it,  whose  corresponding  extremi- 
ties are  N'  and  S'9  N  and  S1  will  attract  each  other,  as  also  S 
and  JV". 

2.  If  not  disturbed,  the  coil  will  place  itself  lengthwise  in  the 
direction  of  the  magnetic  meridian,  with  the  extremity  N  toward 
the  north,  and  S  toward  the  south. 

3.  If  a  bar  of  iron  be  placed  within  it,  the  bar  will  become  a 
magnet,  having  its  north  pole  at  N,  and  its  south  pole  at  S. 

If  a  right-hand  helix  had  been  employed,  all  these  phenom- 
ena would  have  been  reversed. 

466.  Ampere's  Theory  of  Magnetism. — In  these  experi- 
ments a  coil  is  found  to  act  the  same  as  a  magnet  whose  north 
and  south  poles  are  at  N  and  S  respectively.    We  therefore  de- 
duce the  following : 

1.  A  helix  traversed  by  a  galvanic  current  is  a  magnet  the 
position  of  whose  poles  depends  on  the  direction  of  the  current. 

2.  Conversely,  a  magnet,  like  a  coil,  may  be  conceived  to  owe 
its  magnetic  properties  to  currents  of  electricity  which  traverse  it. 

This  is  the  theory  of  Ampere,  and  is  the  one  generally  received, 
notwithstanding  some  objections  to  it. 

In  the  helix  a  single  current  is  present.     But  in  a  magnet  we 


282 


DYNAMICAL    ELECTRICITY. 


FIG.  249. 


must  conceive  of  an  infinite  number  of  currents,  the  circuit  of 
each  being  confined  to  an  individual  molecule.    Pig.  248  repre- 
sents a  magnet  accord- 
ing to  this  theory,  and  FlG  °^- 
N  and  S  (Fig.  249) 
show  the    extremities 
of  the  north  and  south 

poles  on  a  larger  scale.  The  arrows  on  the  convex  surface  show 
the  general  direction  of  all  the  currents — that  is,  of  those  portions 
of  them  nearest  the  surface, 
where  magnetism  is  in  fact 
developed — and  may  there- 
fore represent  them  all. 

Since  S  is  the  south  pole 
of  the  magnet,  as  supposed 
to  be  seen  by  an  observer 
looking  at  it  in  the  direc- 
tion of  its  axis,  it  follows 
that  when  a  magnet  is  in 
its  normal  position,  that 
is,  with  its  north  pole  point- 

ing  northzvard,  its  currents  circulate  from  west  over  to  east,  and 
therefore  from  left  over  to  right  if  the  observer  is  also  looking 
northward.  In  like  manner,  it  is  evident  that  to  a  person  looking 
along  the  length  of  a  magnet,  from  its  north  toward  its  south  pole, 
the  currents  circulate  from  the  right  over  'to  the  left. 

These  supposed  currents  of  the  magnet  are  so  small  that  we 
cannot  take  cognizance  of  them  directly.  But  on  the  basis  of 
Ampere's  theory,  we  may  substitute  for  them  the  large  and  man- 
ageable current  of  a  helix.  Then,  by  determining  experimentally 
the  causes  of  magnetic  phenomena  in  the  case  of  the  latter,  we 
may  assign  the  same  causes  to  like  phenomena  of  the  magnet. 


FIG.  250. 


467.  Mutual  Action  of  Currents. — 

1.  If  galvanic  currents  flow  through  parallel  wires  in  the 
direction,  they  attract  each  other ;  if  in  opposite  directions, 
repel  each  other.  These  effects  are  shown  in  Fig.  250,  where 
A'  Br,  turn  toward  each  other, 
while  C  D,  Cr  D',  turn  away 
from  each  other. 

Hence,  when  a  current  flows 
through  a  loose  and  flexible 
helix,  each  turn  of  the  coil  at- 
tracts the  next,  since  the  current 
moves  in  the  same  direction 


same 
they 
AB, 


CURRENTS    AND    MAGNETS. 


283 


through  them  all.  In  this  waj-,  a  coil  suspended  above  a  cup  of 
mercury,  so  as  to  just  dip  into  the  fluid,  will  vibrate  up  and  down 
as  long  as  a  current  is  supplied.  The  weight  of  the  helix  causes 
its  extremity  to  dip  into  the  mercury  below  it ;  this  closes  the 
circuit,  the  current  flows  through  it,  the  spirals  attract  each  other, 
and  lift  the  end  out  of  the  mercury ;  this  breaks  the  circuit,  and  it 
falls  again,  and  thus  the  movement  is  continued. 

2.  If  currents  flow  through  two  wires  near  each  other,  which 
are  free  to  change  their  directions,  the  wires  tend  to  become  paral- 
lel to  each  other,  with  the  currents  flowing  in  the  same  direction. 
Thus,  two  circular  wires,  free  to  revolve  about  vertical  axes,  when 
currents  flow  through  them,  place  themselves  by  mutual  attrac- 
tions in  parallel  planes,  as  in  Fig.  251,  or  in  the  same  plane,  as  in 
Fig.  252.  In  the  latter  case,  we  must  consider  the  parts  of  the 
two  circuits  which  are  nearest  to  each  other  as  small  portions  of 
the  dotted  straight  lines,  c  d  and  ef. 


It  appears,  therefore,  that  galvanic  currents,  by  mutual  attrac- 
tions and  repulsions,  tend  to  place  themselves  parallel  to  each  other 
in  such  a  manner  that  thefloiv  is  in  the  same  direction. 

Supposing  the  same  to  hold  true  of  the  molecular  currents  of 
magnets,  this  single  law  will  satisfactorily  account  for  the  phe- 
nomena of  magnetic  polarity. 

In  the  following  articles  these  phenomena  are  considered  in  the 
order  in  which  they  are  mentioned  in  Art.  465. 

468.  Relations  of  Currents  and  Magnets  to  Each  Other 
(1.  Art.  465). — It  should  be  constantly  borne  in  mind  that  when 
the  north  pole  of  a  magnet  turns  toward  a  person,  its  currents  cir- 
culate from  his  right  over  to  his  left. 

1.  When  two  solenoids,  suspended  as  in  Fig.  247,  or  when  a 
solenoid  and  a  magnet,  or  two  magnets,  are  brought  near  each 
other,  poles  of  different  names  attract,  and  those  of  the  same  name 


284 


DYNAMICAL    ELECTRICITY. 


repel.    For,  when  the  magnets  suspended  from  A  and  B  (Fig.  253) 
axe  in  the  same  line,  it  is  seen  that  the  currents  are  parallel  and 


FIG.  253. 


flow  in  the  same  direction  in  all  the  corresponding  parts ;  and  in 
Fig.  254,  where  they  hang  side  by  side,  the  nearer  parts  of  the 


FIG.  255. 


FIG.  256. 


currents  are  parallel  and  flow  in  the  same  direction.  "While  in 
Fig.  255,  where  like  poles  are  contiguous,  the  corresponding  parts 
of  the  currents  flow  in  opposite  directions. 

2.  When  a  magnet  is  suspended  within  a  loop  through  which 
a  current  flows,  if  free  to  move 
it  will  place  itself  at  right  angles 
to  the  plane  of  the  circuit,  with 
the  north  pole  pointing  toward 
a  person,  when  the  current 
passes  from  his  right  over  to 
his  left  (Fig.  256).  Therefore, 
if  the  circuit  is  in  a  horizontal 
plane,  the  magnet  turns  its  north 
pole  downward,  if  the  current 
flows  as  in  Fig.  257,  or  upward 
if  the  current  is  reversed. 


THE    GALVANOMETER. 


285 


3.  When  a  magnet  is  brought  near  a  closed  circuit  wire, 
as  H (Fig.  258),  it  will  place  itself  tangentially  to  a  circle,  x  y  z, 


FIG.  257. 


whose  centre  is  in  the  wire,  and  its  plane  perpendicular  to  it. 
The  part  of  the  wire  nearest  to  the  magnet  may  be  considered  as 
a  small  portion  of  a  loop  around  it,  as  in  Fig.  256.  This  tangen- 
tial relation  is  maintained  on  all  sides  of  the  circuit,  it  being 
everywhere  true  that  when  the  north  pole  is  directed  to  a  person, 
the  current  descends  on  the  left,  as  if  it  had  passed  from  the  right 
over  to  the  left. 

Comparing  Figs.  257  and  258,  it  is  evident  that  the  current 
and  the  magnet  may  change  places  without  disturbing  their  rela- 
tive directions,  it  being  understood  that  the  current  flows  in  the 
same  direction  in  which  the  north  pole  points. 

469.  The  Galvanometer. — Advantage  is  taken  of  the  direc- 
tive influence  of  a  current  on  a  magnet  in  the  construction  of  the 
galvanometer  (Fig.  259).  When  the 
coil  consists  of  many  convolutions  of 
wire,  a  very  feeble  current  passing 
through  will  deflect  the  needle  from  its 
north  and  south  direction,  and  the 
amount  of  deflection  serves  as  a  measure 
of  the  galvanic  force.  Hence  the  name 
of  the  instrument.  To  render  it  still 
more  sensitive,  a  second  smaller  needle, 
with  poles  reversed,  attached  to  the  same 
vertical  wire,  makes  the  first  nearly 
astatic  with  relation  to  the  earth.  In  making  such  a  coil,  the 
wire  must  be  carefully  insulated.  This  is  generally  done  by  wind- 
ing it  with  silk  thread.  In  the  figure,  the  galvanometer  is  repre- 


FIG.  259. 


286  DYNAMICAL    ELECTRICITY. 

sentecl  as  covered  by  a  bell-glass.  The  coil  is  seen  beneath  the 
graduated  circle;  the  deflected  needle  projects  as  a  white  line  from 
within  the  coil,  and  directly  above  it  is  the  needle,  which  nearly 
neutralizes  the  earth's  influence  upon  it. 

470.  Polarity  with  Respect  to  the  Earth  (2.  Art.  465).— 

It  is  believed  that  currents  of  electricity  are  constantly  traversing 
the  earth's  crust,  passing  around  it  from  east  to  west,  and  making 
the  earth  itself  a  magnet,  with  boreal  magnetism  developed  at  the 
north  pole,  and  austral  at  the  south  pole.  Thus  the  earth  may  be 
taken  as  the  standard  magnet,  and  both  it  and  the  currents  around 
it  control  the  polarity  of  the  needle.  For,  as  in  Fig.  260,  in  order 

FIG.  SCO. 


that  the  current  of  the  magnet  may  be  parallel  with  the  adjacent 
terrestrial  current,  and  in  the  same  direction  with  it,  since  the 
latter  passes  from  east  to  west,  the  lower  side  of  the  former  must 
also  pass  from  east  to  west.  But  in  order  that  this  may  be  the 
case,  the  north  pole  of  the  magnet  must  point  northward,  and  this 
it  does  when  free  to  obey  the  directive  influence  of  the  earth. 

At  first  view,  the  earth  currents  from  east  to  west  seem  to  be 
in  the  wrong  direction ;  for  that  is  from  left  over  to  right,  to  a 
person  to  whom  the  north  pole  points.  This,  however,  is  ex- 
plained by  recollecting  that  the  magnetism  of  the  north  pole  of 
the  earth  is  the  same  as  that  of  the  south  pole  of  a  magnet  (Art. 
380).  For  convenience,  that  end  of  a  needle  which  points  north 
is  called  the  north  pole ;  but  by  the  law  of  attraction  between  op- 
posite poles,  it  must  be  unlike  the  north  pole  of  the  earth.  There- 
fore, the  rule  for  the  direction  of  currents  around  a  magnet  must 
be  reversed  when  applied  to  the  earth. 


THERMO-ELECTRICITY.  287 

The  existence  of  currents  traversing  the  earth's  crust  has  been 
variously  accounted  for.  The  strong  analogy  between  them  and 
those  of  thermo-electricity  points  to  the  heat  of  the  sun  as  at  least 
a  very  probable  cause. 

471.  Thermo-Electricity. — Let  a  number  of  bars  of  bis- 
muth (#)  and  antimony  (a)  be  soldered  together  as  in  Fig.  261. 
Now  if  the  flame  of  a  candle  be  carried 

around  so  as  to  warm  the  outer  joints,  a  FlG- 

current  of  electricity  will  pass  through 
the  circuit  from  left  to  right,  and  will  in- 
fluence a  needle  near  it  just  as  any  other 
current  would  do,  flowing  in  the  same  di- 
rection. Furthermore,  it  is  only  while  the 
metals  are  unequally  heated  that  the  cur- 
rent flows.  We  may  therefore  suppose  that 
the  terrestrial  current  may  be  caused,  in 
part  at  least,  by  the  unequal  heating  of 

the  heterogeneous  substances  composing  the  earth's  crust,  as  the 
sun's  heat  is  alternately  poured  upon  and  withdrawn  from  them 
once  in  every  diurnal  revolution. 

472.  Magnetic  Induction  by  Currents  (3.  Art.  465). — 

Ampere  accounted  for  the  phenomena  of  magnetic  induction  by 
supposing  that  galvanic  currents  circulate  through  the  molecules 
of  all  bodies,  but  in  different  directions,  so  that  they  mutually 
neutralize  each  other.  That  in  a  few  substances,  such  as  steel  and 
iron,  it  is  possible  to  control  these  currents  and  cause  them  all  to 
flow  in  the  same  direction ;  and  that  when  this  is  done,  the  phe- 
nomena of  polarity  ensue. 

Supposing  this  to  be  the  correct  explanation,  the  effect  of  a 
galvanic  current  (and  in  fact  of  any  method  of  magnetizing)  is 
simply,  by  repulsion  and  attraction,  to  produce  uniformity  of 
direction  among  these  magnetic  currents. 

473.  The  Permanent  and  Temporary  Magnet— When  a 
current  of  sufficient  strength  is  passed  around  a  bar  of  well-tem- 
pered steel,  a  permanent  magnet  of  considerable  power  may  be 
obtained. 

With  soft  iron,  the  result  is  a  temporary  magnet,  which  retains 
its  magnetic  properties  only  while  the  current  is  in  motion.  In 
either  case  the  poles  are  always  in  the  position  which  those  of  a 
needle  would  voluntarily  assume  if  placed  in  the  same  relation  to 
the  current. 

474.  The  U-Magnet. — Let  a  piece  of  soft  iron,  in  the  form 
of  a  horseshoe  or  the  letter  U  (Fig.  262),  be  wound  with  a  coil  of 


288 


DYNAMICAL    ELECTRICITY. 


FIG.  262. 


insulated  copper  wire  whose  extremities,  W  and  w,  are  dipped  in 
cups  of  mercury,  in  which  are  also  dipped  the  electrodes  +  and  — 
of  a  battery.  When  all  the  wires  are 
in  metallic  communication,  the  cir- 
cuit is  closed,  and  the  current  pass- 
ing around  the  iron  makes  it  a  mag- 
net ;  and  since  to  a  person  looking 
along  the  length  of  the  helix  the 
current  passes  from  right  over  to 
left,  the  north  pole  is  at  .A7",  and  the 
south  pole  at  S.  As  soon  as  the 
circuit  is  broken  by  lifting  out  of 
the  mercury  any  one  of  the  wires, 
the  weight  which  was  previously  sus- 
tained will  fall,  showing  that  the 
iron  is  no  longer  a  magnet. 

475.  Helices. — The  form  of  coil  or  helix  generally  employed 
is  shown  in  Fig.  263.  Many  hun- 
dreds or  even  thousands  of  feet  of 
insulated  wire  are  wound  around  two 
bobbins,  and  through  the  centre  of 
each  passes  a  branch  of  the  U-shaped 
iron ;  or,  more  frequently  the  central 
cores  of  iron  are  separate  pieces, 
joined  by  a  third  one  across  two  of 
the  ends,  and  thus  a  U-magnet  of 
modified  form  is  obtained.  By  em- 
ploying a  fine  wire  coiled  many  times 
around  the  bobbins,  a  magnet  of  very 
great  power  may  be  formed,  consider- 
ing the  weakness  of  the  battery  which 
furnishes  the  current.  A  magnet 
formed  by  the  use  of  a  small  Bunsen 
cell  has  been  known  to  lift  five  hun- 
dred pounds,  and  with  twenty  Grove  cells  can  be  made  to  sustain 
a  weight  of  three  tons. 


FIG.  263. 


INDUCED    CURRENTS. 


289 


CHAPTER    III. 

INDUCED    CURRENTS. 

I.  CURRENTS  INDUCED  BY  CURRENTS. 

476.  Experiment. — Place  a  copper  wire,  a  I  (Fig.  264),  near 
p  n,  the  circuit  wire  of  a  battery.  The  current  flowing  through 
p  n  acts  inductively  on  a  b,  decomposing  its  natural  electricity. 
According  to  the  general  law  of  induction,  the  positive  electricity  of 

FIG.  264. 


a  I  is  attracted  in  the  direction  of  n,  and  the  negative  in  the  direc- 
tion of  p.    In  this  experiment  let  the  following  facts  be  noticed : 

(1)  The  decomposition  of  the  natural  electricity  of  at  occurs 
at  the  instant  of  dosing  the  circuit  p  n. 

(2)  While  the  circuit    remains    dosed,  the  current  passing 
through  it  does  not  induce  a  current  through  a  b.    There  is  prob- 
ably, however,  an  accumulation  of  positive  electricity  in  the  direc- 
tion of  b,  and  of  negative  in  the  direction  of  a ;  for 

(3)  When  the  circuit  is  broken,  the  natural  electrical  equilib- 
rium of  a  b  is  instantly  restored,  and  no  further  signs  of  a  current 
can  be  detected  until  the  circuit  is  again  closed. 

The  circuit  is  conveniently  closed  by  dipping  the  end  of  the 
wire,  e  or  d,  in  the  cup  of  mercury  m,  and  broken  by  removing  it 
from  the  cup. 

477.  The  Induced  and  Inducing  Currents. — The  sudden 
decomposition  of  the  natural  electricity  mentioned  in  (1)  of  the 
preceding  Article  involves  a  momentary  flow  of  the  two  electrici- 
ties in  opposite  directions ;  that  is,  a  current  is  made  to  traverse 
the  wire  from  a  to  b,  that  being  the  direction  of  the  flow  of  the 
positive  fluid  (Art.  457).  And  the  restoration  of  equilibrium 
mentioned  in  (3)  involves  a  reversal  of  this  flow ;  that  is,  it  pro- 
duces a  current  which  passes  from  b  to  a. 
19 


290  DYNAMICAL    ELECTRICITY. 

These  two  currents  in  a  b  are  called  induced  currents  ;  and  the 
one  in  p  n,  to  which  they  owe  their  origin,  is  called  the  inducing 
current.  The  presence,  direction,  and  duration  of  the  induced 
currents  are  indicated  by  the  galvanometer  g. 

The  terms  flow,  current,  accumulation  of  electricity,  &c.,  when 
applied  to  the  whole  wire,  have  no  significance  except  as  they  in- 
dicate the  resultants  of  those  electrical  disturbances  which  are 
probably  confined  to  the  individual  molecules  of  the  wire. 

478.  Characteristics  of  Induced  Currents.— It  is  obvious 
that  induced  currents  differ  materially  from  the  current  of  a  bat- 
tery which  is  uniform  in  direction  and  constant  in  intensity  for 
an  appreciable  length  of  time.    The  following  are  the  distinctive 
features  of  induced  currents : 

(1)  Induced  currents  are  instantaneous. 

(2)  They  result  from  interruptions  of  the  inducing  current. 

(3)  On  closing  the  circuit,  the  direction  of  the  resulting  in- 
duced current  is  opposite  to  that  of  the  inducing  current. 

(4)  On  breaking  the  circuit,  the  induced  and  inducing  currents 
are  in  the  same  direction. 

479.  Inducing  and  Induced  Currents  in  one  Wire. — 

We  have  thus  far  considered  only  the  inductive  influence  of  the 
current  on  a  wire  exterior  to  its  circuit.  But  the  circuit-wire  p  n 
itself  possesses  its  share  of  natural  electricity,  as  well  as  a  b. 

This  is  believed  to  exist  independently  of  the  galvanic  current 
passing  through  it,  and  to  be  decomposed  by  that  current.  In 
order,  therefore,  to  produce  the  preceding  results  with  a  single 
wire,  let  the  circuit-wire  be 

coiled  as  in  Fig.  265.    Each          t _      FIG.  265^ 

spire  is  now  acted  upon  induc- 
tively by  the  galvanic  current 
passing  through  the  adjacent 
spires  in  the  manner  already 
described  for  separate  wires. 
In  addition  to  this  mutual  in- 
ductive influence  of  the  several 
spires  on  each  other,  it  is  prob- 
able that  the  natural  electricity  of  every  portion  of  the  wire  is  still 
further  decomposed  by  the  galvanic  current  passing  through  it 
For  it  is  a  noticeable  fact  that  when  a  very  long  circuit-wire  is 
employed,  induced  currents  are  obtained  even  though  it  be  so 
nearly  straight  that  no  one  portion  can  act  inductively  on  another. 

The  result  of  these  several  inductive  actions  is  that  when  the 
circuit  is  closed  and  broken,  regular  induced  currents  are  gen- 
erated in  it.  And  since  these  coexist  for  an  instant  of  time  with 


INDUCED    CURRENTS.  291 

the  inducing  current,  and  pass  through  the  same  electrodes  with  it, 
it  follows — 

(1)  That  when  the  circuit  is  closed,  the  inducing  current  is 
partially  neutralized,  and  has  its  intensity  diminished  hy  the  in- 
duced current  which  flows  in  a  direction  contrary  to  its  own ; 
and 

(2)  That  when  it  is  opened,  the  induced  current  having  now 
the  same  direction  as  the  inducing  current,  reinforces  it  and  aug- 
ments its  intensity. 

480.  Mode  of  Naming  Circuits  and  Currents. — The  phe- 
nomena of  induced  currents  were  discovered  by  Faraday  in  1832, 
and  to  him  we  owe  the  foregoing  explanation  of  them.     The  fol- 
lowing terms  now  in  use  were  also  introduced  by  him : 

The  inducing  current  is  called  the  primary  current,  and  the 
wire  it  traverses  the  primary  wire.  Currents  induced  in  the  pri- 
mary wire  are  called  extra  currents ;  the  one  obtained  on  closing 
the  circuit  is  the  inverse  extra  current ;  the  one  on  opening  it  is 
the  direct  extra  current  (Art.  478,  3,  4). 

A  wire  exterior  to  the  primary,  as  a  b  in  Fig.  264,  is  a  sec- 
ondary wire,  and  the  currents  induced  in  it  are  secondary  cur- 
rents. 

481.  Currents    Induced    in    Coils. — Instead  of  straight 
wires  or  loose  spirals,  compact  coils  of  carefully  insulated  wire  are 
employed.    Thus  all  parts  of  the  wire  are  brought  much  nearer  to 
each  other,  and  the  inductive  influence  is  far  more  energetic.    In- 
deed, without  a  coil,  the  presence  of  induced  currents  can  gener- 
ally be  detected  only  with  a  delicate  galvanometer.    The  following 
experiments  show  the  effects  of  coils : 

(1)  Around  a  hollow  wooden  bobbin,  5  (Fig.  266),  coil  about 
100  feet  of  No.  16  insulated  copper  wire.  Let  this  be  made  a  part 
of  the  circuit  of  a  battery,  as  shown  in  the 
figure.  This  circuit  is  of  course  closed  ^  FlG- 
when  m  and  n  touch  each  other.  Now  if 
m  and  n  be  held  one  in  each  hand  and 
then  separated,  the  body  of  the  operator 
becomes  a  part  of  the  circuit,  and  the  pri- 
mary current,  not  having  sufficient  inten- 
sity to  pass  through  it,  ceases.  But  the 
direct  extra  current  passes  through,  producing  a  shock.  When 
the  wires  are  brought  together  again,  the  primary  and  inverse 
extra  currents  pass  through  the  metallic  circuit,  and  no  shock  is 
felt. 

The  more  rapid  the  rate  at  which  m  and  n  are  brought  to- 
gether and  separated,  the  more  decided  are  the  results  obtained. 


292 


DYNAMICAL    ELECTRICITY. 


FIG.  267. 


To  produce  the  most  marked  effect,  attach  a  coarse  file  to  one  end, 
as  m,  and  hold  it  in  one  hand  while  n  is  drawn  rapidly  over  the 
ridges  of  its  surface  with  the  other. 

(2)  Fig.  267  represents  the  same  coil  as  Fig.  266,  with  the  ad- 
dition of  a  bundle  of  soft  iron  wires,  w,  inserted 

in  the  hollow  bobbin. 

When  the  circuit  is  closed,  these  wires  are 
magnetized — that  is,  the  Amp£rean  currents  sup- 
posed to  reside  in  them  are  made  to  circulate  in 
the  same  direction  as  the  battery  current  (Art. 
472). 

And  since  the  appearance  and  disappearance 
of  these  magnetic  currents  are  simultaneous  with 
the  appearance  and  disappearance  of  the  primary 
current,  they  augment  the  effects  of  the  latter,  and  the  resulting 
extra  currents  are  of  greater  intensity. 

The  effect  of  soft  iron  in  the  primary  coil  is  an  observed  fact, 
and  the  above  is  the  way  in  which  Faraday  accounted  for  it  on 
the  basis  of  Ampere's  theory. 

(3)  Let  the  primary  coil  and  bundle  of  wires  of  the  preceding 
figure  be  placed  within  a  secondary  coil,  d  (Fig.  268),  from  which 
it  is  carefully  insulated.    This 

secondary  coil  should  be  made 
of  wire  much  greater  in 
length  and  smaller  in  diam- 
eter than  that  of  which  the 
primary  coil  is  made.  For 
instance,  let  it  consist  of  1500 
feet  of  No.  35  insulated  cop- 
per wire.  When  the  ends  of 
this  wire,  li,  h',  are  held  one 
in  each  hand,  every  time  the 
primary  circuit  is  interrupted, 
a  secondary  current  traverses 
the  secondary  circuit  of  which  the  person  forms  a  part.  The  re- 
sulting shocks  will  be  quite  appreciable,  though  the  primary  cur- 
rent be  produced  by  only  a  single  small  cell. 

As  in  the  first  experiment,  the  effect  on  the  person  will  become 
more  marked  as  the  interruptions  increase  in  frequency. 

In  the  third  experiment,  the  magnetic  currents  of  the  iron  core 
add  their  inductive  influence,  as  already  explained,  to  that  of  the 
primary  current,  thus  increasing  the  intensity  of  the  secondary 
currents. 

The  effect  of  the  extra  currents  also  should  not  be  overlooked. 
As  these  traverse  the  primary  coil,  alternating  with  each  other  in 


FIG.  268. 


RUHMKORFF'S    COIL. 


293 


FIG.  269. 


direction,  they  materially  modify  the  effects  of  the  primary  and 
magnetic  currents,  which  are  uniform  in  direction. 

482.  Ruhmkorff's  Coil. — The  celebrated  Ruhmkorff  coil 
(Fig.  269)  is  not  essentially  different  from  the  one  just  described, 
except  in  having  (1)  an  ar- 
rangement for  producing  a 
continued  and  rapid  succes- 
sion of  interruptions  in  the 
primary  current,  (2)  a  com- 
mutator, or  key,  by  which 
when  desired  the  primary 
current  may  be  stopped  or 
its  direction  reversed,  and 
(3)  a  condenser  to  neutralize 
the  effects  of  the  extra  cur- 
rents. The  condenser  con- 
sists of  sheets  of  tin-foil  in- 
sulated by  oiled  silk.  They 
are  placed  out  of  sight,  in 
the  base  of  the  apparatus, 
and  are  so  connected  with 

the  primary  wire  that  the  extra  currents  pass  into  them.  Owing 
to  this  diversion  of  these  interfering  currents  the  efficiency  of  the 
coil  is  very  much  increased. 

483.  Power  of  the  Ruhmkorff  Coil.— The  efficiency  of  a 
Ruhmkorff  coil  depends  largely  on  complete  insulation;  and,  in 
different  coils,  varies  greatly  with  the  length  and  fineness  of  the 
secondary  ivire. 

To  secure  insulation,  the  wires  are  (as  usual)  wound  with  silk 
thread,  then  each  individual  coil  around  the  axis,  is  separated  from 
the  succeeding  one  by  a  layer  of  melted  shellac,  and  lastly  a  cylin- 
der of  glass  is  placed  between  the  primary  and  secondary  -coils. 

With  regard  to  the  secondary  coil,  one  of  the  largest  size  some- 
times contains  sixty  miles  of  the  finest  copper  wire.  With  such 
an  apparatus,  though  the  primary  current  be  produced  by  only 
two  or  three  Bunsen  cells,  the  secondary  currents  are  of  such  in- 
tensity that  sparks  eighteen  inches  in  length,  and  of  great  bril- 
liancy, may  be  obtained ;  and  indeed,  all  the  tension  effects  of  a 
large  electrical  machine,  as  well  as  the  quantity  effects  of  a  power- 
ful galvanic  battery,  may  be  reproduced. 

Great  care  should  be  taken  in  handling  an  induction  coil  of 
this  size,  for  the  shock  resulting  from  its  discharge  through  the 
body  would  be  dangerous,  and  might  possibly  prove  fatal. 


294 


DYNAMICAL    ELECTRICITY. 


FIG.  270. 


484.  One  Coil  moved  into,  and  out  of,  another.— In  all 

that  has  preceded,  the  interruptions  of  the  primary  current  have 
been  supposed  to  take  place  instantaneously.  If  these  interrup- 
tions are  gradual,  the  resulting  induced  currents  remain  the  same 
in  direction  as  before,  but  vary  in  intensity  and  duration. 

Thus,  if  the  primary  coil  c  (Fig.  270)  be  made  to  fit  loosely  in 
the  secondary  coil  d,  and  then  be  moved  up  and  down  (the  pri- 
mary circuit  remaining 
closed),  it  will  be  found — 

(1.)  That  each  inser- 
tion and  removal  of  it  cor- 
responds, the  one  to  a 
gradual  closing,  the  other 
to  a  gradual  opening  of  the 
primary  circuit — the  result 
of  the  former  being  an  in- 
verse secondary  current,  of 
the  latter  a  direct  seconda- 
ry current. 

And  since  a  continuous 
motion  of  the  primary  coil 
produces  a  continuous  se- 
ries of  instantaneous  sec- 
ondary currents  with  no 
appreciable  interval  between  them,  it  will  be  found — 

(2.)  That  the  secondary  currents  are  continuous  in  effect  as 
long  as  the  motion  of  the  primary  coil  is  continuous; 

(3.)  That  their  intensity  varies  with  the  rate  of  motion  of  the 
primary  coil,  diminishing  or  increasing  as  that  is  moved  slowly  or 
rapidly ;  from  which  it  follows — 

(4.)  That  they  cease  whenever  the  primary  coil  is  brought  to  a 
state  of  rest  in  any  position. 

485.  Changes  of  Intensity  in  the  Primary  Current. — All 

the  results  just  mentioned  may  be  obtained  if,  instead  of  changing 
the  position  of  the  primary  coil,  as  above,  it  remain  at  rest  while 
a  corresponding  series  of  variations  be  produced  in  the  primary 
current, — an  increase  of  intensity  in  that  corresponding  to  an 
insertion  of  the  coil,  and  a  decrease  to  a  removal  of  it. 


II.    CUREEOTS  INDUCED  BY  MAGNETS. 

486.  Magneto-electricity. — Faraday  reasoned  that  if  cur- 
rents could  induce  magnetism,  a  magnet  ought  to  induce  currents. 


MAGNETO-ELECTRICITY. 


295 


FIG.  271. 


This  he  found  to  be  the  case,  and  thus  discovered  a  new  branch 
of  physical  science,  to  which  he  gave  the  name  of  magneto-elec- 
tricity. 

If  a  magnet  be  used  instead  of  the  primary  coil  in  Fig.  270,  all 
the  phenomena  mentioned  in  Art.  484  may  be  reproduced. 

Thus,  with  the  coil  and  magnet  in 
Fig.  271,  we  obtain  the  following  re- 
sults : 

(1.)  When  the  magnet  is  alternately 
inserted  in  and  withdrawn  from  the  coil, 
the  latter  is  traversed  by  induced  cur- 
rents alternating  with  each  other  in  di- 
rection. 

(2.)  These  currents  are  continuous 
while  the  magnet  is  in  motion. 

(3.)  Their  intensity  diminishes  or 
increases  as  the  magnet  moves  slowly  or 
rapidly. 

(4.)  They  cease  when  the  motion  of  the  magnet  ceases. 

487.  Explanation  of  the  foregoing  Phenomena. — On  the 

basis  of  Ampere's  theory,  the  correspondence  of  these  phenomena 
with  those  of  Art.  484  can  readily  be  accounted  for.  For  the  mag- 
net may  be  considered  a  true  primary  coil,  its  magnetic  currents 
corresponding  to  the  primary  current  in  Fig.  270.  With  regard 
to  them,  the  induced  currents  are  regular  inverse  and  direct  sec- 
ondaries; for  in  any  given  case  they  will  be  found  to  have  the 
same  direction  as  those  induced  by  a  primary  current  whose  di- 
rection corresponds  with,  the  supposed  direction  of  the  magnetic 
currents. 

It  will  be  seen  at  once  what  a  strong  argument  is  here  fur- 
nished in  favor  of  Ampere's 
theory. 

*  488.  An  Iron  Core, 
changing  its  Magnetic 
Intensity.  —  Replace  the 
magnet  (Fig.  271)  by  a  bar 
of  soft  iron  inserted  in  the 
coil,  and  let  a  magnet  be 
alternately  brought  near 
this,  and  removed  from  it, 
as  in  Fig.  272.  The  same 
results  will  be  obtained  as 
in  the  preceding  series  of 
experiments.  The  proxim- 


FIG.  272. 


296  DYNAMICAL    ELECTRICITY. 

ity  of  the  magnet  induces  magnetism  in  the  soft  iron,  and  its 
motions  to  and  fro  produce  variations  in  this  induced  magnetism 
corresponding  precisely  with  the  varying  intensity  of  the  primary 
current  mentioned  in  Art.  485,  and,  as  might  be  expected,  the 
results  are  the  same. 

In  this  experiment,  it  is  obviously  immaterial  whether  the  coil 
be  at  rest  and  the  magnet  be  moved,  or  the  magnet  be  at  rest  and 
the  coil  be  moved.  The  latter  method  is  adopted  in  some  mag- 
neto-electric machines. 

489.  Clarke's  Magneto-electric  Machine.— In  front  of 
the  poles  of  the  U-magnet,  A  (Fig.  273),  is  revolved  the  armature, 

FIG.  273. 


consisting  of  the  two  lolUns,  B,  B',  which  are  coils  of  fine  wire 
with  cores  of  soft  iron.  These  cores  are  joined  to  each  other  and 
to  the  axis  of  rotation  by  the  bar  of  soft  iron,  V,  and  motion  is 
communicated  by  the  multiplying  wheel  and  band  at  W. 

As  One  of  the  bobbins  passes  before  a  north  pole  while  the 
other  is  passing  before  a  south  pole,  the  resulting  induced  cur- 
rents are  relatively  of  contrary  directions ;  but  as  one  of  the  coils 
is  always  right-handed  and  the  other  left-handed,  the  currents 
passing  through  them  at  any  given  instant  have  the  same  abso- 
lute direction,  so  that  the  two  coils  act  as  one. 


THE    COMMUTATOR. 


297 


FIG.  274. 


Fig.  274  shows  the  poles  of  the  fixed  magnet,  and  the  direction 
in  which  the  armature  revolves.  The  maximum  magnetization 
of  the  soft  iron  cores  occurs  when  the  bobbins  are  directly  in  front 
of  N  and  S.  While  they  move  through 
the  first  and  third  quadrants  they  are 
losing  their  magnetism,  and  while  mov- 
ing through  the  second  and  fourth  they 
are  acquiring  that  of  the  contrary  kind. 
The  resulting  induced  currents  will  thus 
be  direct  and  inverse  to  contrary  kinds 
of  magnetism,  and  will  therefore  have 
the  same  absolute  direction.  But  as  the 
bobbins  pass  from  the  second  quadrant 
to  the  third,  and  from  the  fourth  to  the 
first,  they  lose  the  magnetism  just  ac- 
quired, and  the  induced  currents  change 
from  inverse  to  direct  with  reference  to 
the  same  kind  of  magnetism,  and  there- 
fore become  reversed  in  absolute  direc- 
tion. 

Hence  the  semi-revolutions  of  the 
armature  on  opposite  sides  of  a  line 

joining  the  poles  of  the  permanent  magnet  produce  currents  of 
contrary  directions. 

490.  The  Commutator.— Fig.  275  is  an  enlarged  view  of  the 
outer  end  of  the  axis  (Fig.  273),  and  shows  the  commutator,  or 

FIG.  275. 


V 


' 


arrangement  by  means  of  which  the  contrary  currents  just  men- 
tioned are  made  to  furnish  one — or  rather  a  series  of  currents — 
flowing  in  the  same  direction. 

Two  pieces  of  brass,  m  and  n,  are  insulated  from  each 
other  by  being  fastened  to  an  ivory  ring,  i,  around  the  axis.  They 
are  so  connected  with  the  wires  of  the  coils  that  at  any  given  in- 
stant both  coils  present  to  m  one  kind  of  polarity,  and  to  n  the 


298  DYNAMICAL    ELECTRICITY. 

other ;  that  is,  they  are  made  the  poles  of  the  two  coils  acting  as 
one.  Against  m  and  n  press  two  springs,  1)  and  c,  which  are 
brought  into  communication  with  each  other  through  the  plates 
P  and  P',  and  the  handles  h  and  h',  whenever  the  latter  are  joined. 
When  b  and  c  press  against  m  and  n  respectively,  it  will  be  seen 
that  the  circuit  is  complete.  Let  us  suppose  that  the  induced  cur- 
rent is  passing  through  it  in  the  direction  indicated  by  the  arrows. 
When  the  armature  has  revolved  through  180°  from  its  present 
position,  m  and  n  will  have  changed  places — but  they  will  also  have 
changed  polarities  (Art.  489).  Therefore  n  presents  to  b  the  same 
polarity  which  m  did,  and  hence  there  is  no  change  in  the  direc- 
tion of  the  current  through  b  and  c. 

491.  Effects  of  Rapid  Revolution. — The  intensity  of  the 
induced  currents  of  this  machine,  as  also  the  rapidity  with  which 
they  succeed  each  other,  is  regulated  by  the  rate  of  revolution  of 
the  armature.    When  this  is  rapidly  revolved,  they  produce  all  the 
effects  of  a  single  voltaic  current,  so  that  the  apparatus  may  be 
used  as  a  galvanic  battery  with  h  and  h'  for  its  electrodes.    At  the 
same  time  its  physiological  effects  are  most  remarkable,  the  shocks 
becoming  unendurable  when  it  is  revolved  with  great  rapidity. 
The  shocks  are  more  powerful  when  a  third  spring,  a  (Fig.  274), 
is  attached  to  the  plate  P'9  near  c. 

492.  Large  Machines. — In  large  magneto-electric  machines 
of  this  kind,  increased  efficiency  is  obtained  in  two  ways : 

First)  by  multiplication  of  magnets.  In  Nollet's  machine,  con- 
structed in  1850,  192  magnetized  steel  plates  are  so  combined  as 
to  make  40  powerful  U-magnets.  These  are  arranged  in  eight 
rows  around  the  circumference  of  a  large  iron  frame  inside  of 
which  revolve  sixty-four  bobbins. 

Second,  by  multiplication  of  currents.  In  Wild's  machine,  con- 
structed on  this  plan,  the  induced  currents  first  obtained,  instead 
of  being  directly  utilized,  are  passed  through  the  coils  of  a  large 
electro-magnet.  Before  the  poles  of  this  a  second  armature  re- 
volves, and  the  resulting  induced  currents  are  far  more  powerful 
than  the  first.  These  may  in  turn  be  made  to  magnetize  a  second 
electro-magnet  before  the  poles  of  which  a  third  armature  re- 
volves, &c. 

Currents  of  very  great  intensity  may  be  obtained  from  either 
of  these  machines.  The  motive  power  employed  is  generally  a 
steam-engine  of  from  one  to  fifteen-horse  power. 


ELECTROLYSIS. 


299 


CHAPTER   IY. 

PRACTICAL    APPLICATIONS. 

493.  Classification. — The  applications  of  Galvanic  electricity 
in  the  arts  and  sciences,  as  well  as  in  the  affairs  of  every  day  life, 
are  eminently  practical.  They  may  be  classified  according  to  the 
way  in  which  are  utilized  those  molecular  forces  whose  resultant 
is  known  as  the  current. 

It  may  be  stated  in  general  that  these  applications  are  made 
either  within  the  circuit,  or  exterior  to  it. 

Within  the  circuit.  Here  the  electrical  force  is  employed  (I) 
directly,  or  (II)  by  being  first  made  to  produce  the  effects  light 
and  heat,  which  are  then  applied  as  desired. 

Without  the  circuit.  Here  it  is  employed  indirectly  (III)  by 
being  made  to  reappear  as  mechanical  motion  through  the  inter- 
vention of  the  kindred  force,  magnetism. 

Examples  will  be  given  of  each. 


FIG.  276 


I.  DIKECT  APPLICATIONS  OF  THE  CUKKEOT. 

494.  Electrolysis. — When  a  current  is  passed  through  a  bi- 
nar}r  compound  (i*e.,  one  containing  two  elements),  the  compound 
is  decomposed,  one  of  its  elements  appearing  at  the  positive  elec- 
trode, the  other  at  the  negative. 

For  instance,  water,  consisting  of  the  two  gases  oxygen  and 
hydrogen,  is  thus  decomposed. 

In  the  bottom  of  the  dish  D  (Fig.  276),  partly  filled  with  water, 
are  fastened  p  and  n,  the 
platinum  electrodes  of  a  bat- 
tery. Over  these  are  placed 
two  tubes,  0  and  H,  full  of 
water.  On  closing  the  cir- 
cuit, oxygen  rises  from  p 
into  0,  and  hydrogen  from 
n  into  H. 

Electrolysis  is  of  the  ut- 
most importance  in  chemis- 
try. Thus,  the  preceding 
experiment  gives  a  correct 
analysis  of  water,  and  if  oxygen  had  been  previously  unknown, 


300  DYNAMICAL    ELECTRICITY. 

would  have  been  the  means  of  its  discovery.    In  this  way  were 
discovered  several  of  the  metallic  elements. 

No  less  important  are  its  applications  in  the  arts.  For  when  a 
solution  of  a  metallic  salt  is  subjected  to  the  action  of  the  current, 
it  is  decomposed  and  a  permanent  film  of  the  metal  is  deposited 
on  any  suitable  material  placed  so  as  to  receive  it.  The  process  is 
then  called  electro-metallurgy,  or  electro-plating. 

495.  Electro-plating. — The  bath  (Fig.  277)  contains  a  satu- 
rated solution  of  blue  vitriol  (sulphate  of  copper).    In  this  is  sus- 
pended by  wires  from  the  metallic  rod  D  a  plate  of  copper  C,  and 
from  B  (also  metallic)  the 

cast  of  a  medal  m,  which  FlG> 

is  to  be  coated  with  copper. 
Connect  D  with  the  posi- 
tive electrode  of  a  battery 
and  B  with  the  negative. 
The  current  passing 
through  the  solution  re- 
moves from  it  particles  of 
copper  and  deposits  them 
on  m.  Those  taken  from 
the  liquid  are  replaced  by  others  taken  from  C,  which  is  thus 
gradually  wasted  away,  and  the  solution  is  kept  saturated. 

If  the  bath  contains  a  solution  of  gold,  and  C  is  replaced  by 
a  piece  of  gold  and  m  by  a  silver  cup,  the  cup  will  be  electro-gilded. 
Electro-silvering  is  an  analogous  process. 

To'  produce  in  any  case  a  firm  and  even  coating,  the  process 
must  be  allowed  to  proceed  slowly  by  the  employment  of  a  weak 
current.  On  a  small  scale  a  single  cell  is  sufficient.  In  large  es- 
tablishments a  magneto-electrical  machine  turned  by  steam  has 
been  successfully  and  economically  used. 

496.  Electrotyping. — By  taking  proper  precautions,  the  cop- 
per film  deposited  on  m  may  be  removed,  and  its  surface  will  be 
found  to  present  an  exact  fac-simile  of  the  medal  of  which  m  is  an 
impression.    Therefore  if  m  is  an  impression  of  the  type  from 
which  a  page  is  printed,  when  the  copper  has  been  removed  and 
stiffened  by  melted  lead  (or  some  alloy)  poured  over  its  under  sur- 
face, it  may  be  used  in  the  printing-press  instead  of  the  type.    It 
is  then  called  an  electro-type  plate,  and  when  not  in  use  may  be 
preserved  indefinitely  for  succeeding  editions,  while  the  type  of 
which  it  is  a  copy  can  be  distributed  and  used  for  other  purposes. 

497.  Medicinal  Applications. — The  shocks  produced  by 
the  passage  of  interrupted  currents  through  the  system  have  al- 


LIGHT    BY    THE    ELECTRIC    CURRENT. 


301 


ready  been  alluded  to.  In  certain  ailments  these  shocks,  when 
properly  applied,  have  been  known  to  produce  beneficial  results. 
On  the  other  hand,  great  injury  has  resulted  from  their  misappli- 
cation. Hence  they  should  be  employed  as  remedies  only  under 
the  direction  of  a  reliable  physician. 

The  familiar  medical  magneto-electrical  machine,  which  comes 
compactly  stored  in  a  box  ten  inches  long  and  about  four  inches 
square  at  the  end,  does  not  differ  essentially  from  Clarke's  (Art. 
489),  except  in  lacking  the  commutator,  so  that  its  currents  pass 
through  the  body  alternating  with  each  other  in  direction. 

II.  APPLICATIONS  OF  ELECTRIC  LIGHT  AND  HEAT. 

498.  Light  by  the  Electric  Current. — The  electric  light 
may  be  advantageously  employed  for  brilliant  illumination  on 
special  occasions ;  also  where  a  strong  penetrating  light  is  needed, 
as  in  light-houses,  or  for  signals  between  ships.     For  exhibiting 
to  an  audience  magnified  images  of  small  objects  (as  with  the  pro- 
jecting microscope)  it  has  no  superior ;  and  to  the  physical  experi- 
menter the  various  colors  it  assumes  on  passing  through  highly 
rarefied  gases  of  different  kinds  are  of  great  interest.    But  as  yet 
it  cannot  compete  with  gas-light  for  ordinary  illumination. 

To  obtain  the  most  brilliant  effects  carbon  electrodes  must  be 
employed,  and  as  these  are  constantly  changing  in  length  they 
must  be  kept  at  a  uniform  distance  apart  by  machinery.     The 
'  flame  is  not  straight,  but  curved,  as  in  Fig.  278,  and  is  called  the 
voltaic  arc.     To  ob- 
tain it,  the  electrodes  Fm-  278- 
must  first  be  made 
to  touch  each  other. 
With  92  Bunsen  ele- 
ments the  light  has 
been  found  to  pos- 
sess more  than  one- 
third  the  intensity  of 
direct  sunlight. 

499.  Heat   by 
the  Electric  Cur- 
rent.— The  heat  as 
well  as  the  light  of 
the  voltaic  arc  is  in- 
tense.   In  the  labo- 
ratory it  is  employed 
to  deflagrate  and  vol- 


302 


DYNAMICAL    ELECTRICITY. 


atilize  refractory  substances.  "When  the  lower  electrode  is  hol- 
lowed out  in  the  form  of  a  cup,  a  piece  of  platinum  (one  of  the 
least  fusible  of  metals)  placed  in  it  is  melted  like  wax  in  a  candle, 
and  a  diamond,  the  hardest  of  known  substances,  is  burnt  to  a 
black  cinder.  To  produce  either  of  these  results,  a  battery  of 
great  power  must  be  used. 

Metals  may  also  be  deflagrated  by  being  made  part  of  the  cir- 
cuit in  the  form  of  very  fine  wires.  They  are  thus  employed  to 
spring  mines  in  time  of  war,  or  in  blasting  rocks.  In  Fig.  279,  J3 
is  a  box  full  of  fulminating  powder,  and  w  is  a  very  fine  platinum 

FIG.  279. 


wire,  about  |  in.  in  length,  fastened  to  p  and  n,  which  are  insu- 
lated copper  wires  extending  to  a  battery  situated  at  any  conve- 
nient distance.  B  and  its  contents  are  the  fuse  which  is  inserted 
in  the  powder  to  be  fired.  When  a  moderately  strong  current 
passes  through  w,  it  is  heated  sufficiently  to  ignite  the  fuse,  and 
the  powder  explodes. 

III.  MECHANICAL  APPLICATIONS. 

500.  Made  through  the  Medium  of  Induced  Magnetism. 

— As  has  been  seen  in  preceding  experiments,  magnetism  may  be 
induced  in  a  piece  of  steel  or  iron  by  the  current  passing  through 
a  circuit  which  is  near  it.  Hence  induced  magnetism  and  its  ap- 
plications are  results  obtained  outside  of  the  circuit. 

This  magnetism  may  be  utilized  directly.  For  compass  and 
galvanometer  needles  are  ordinarily  made  by  placing  a  steel  needle 
in  a  helix  through  which  a  current  is  sent  (Art.  473). 

But  its  most  numerous  and  important  applications  are  in  the 
way  of  mechanical  movements.  All  these  are  modifications  of  the 
simple  rising  and  falling  of  the  armature  of  a  U-magnet,  men- 


tioned  in  Art.  474.  Thus,  the 
armature  A  (Fig.  280)  is  lim- 
ited in  its  fall  by  the  metallic 
base  B,  so  that  it  is  within  the 
influence  of  J/ the  next  time 
that  it  becomes  a  magnet. 
Hence,  when  the  circuit  is 
closed  and  broken  at  n  p,  the 


FIG.  280. 


ELECTRO-MAGNETIC    TELEGRAPH. 


303 


FIG.  281. 


end  A  of  the  lever  L  rises  and  falls.  It  is  evident  that  the  cor- 
responding motions  of  the  end  E  may  be  applied  in  a  variety  of 
ways.  A  lew  of  these  are  described  in  the  following  articles. 

501.  Electro-magnetic  Engine. — E  may  be  attached  to  a 
vertical  arm,  and  that  to  the  crank  of  a  fly-wheel  (Fig.  281),  and 
the  interruptions  of  the  current  may  be  made  automatic  by  con- 
necting p   with   L,  and  n 

with  B.  When  A  rests  on 
B  the  circuit  is  closed ;  M 
becomes  a  magnet,  and  A  is 
attracted  by  it ;  but  as  soon 
as  A  rises  from  B  the  cir- 
cuit is  opened,  M  no  longer 
attracts  it,  and  it  falls  back, 
only  to  close  the  circuit 
again  and  repeat  the  same 
movements  as  before.  The 
tendency  of  this  is  to  pro- 
duce a  rotary  motion  in  the 
fly-wheel,  and  the  apparatus 
involves  the  principle  of  a 
single-acting  engine.  With  a  second  electro-magnet,  and  a  some- 
what different  arrangement  of  parts,  an  actual  double-acting  en- 
gine may  be  constructed. 

Various  forms  of  this  engine  have  been  constructed  and  exhib- 
ited as  curiosities,  or  used  where  expense  was  not  regarded.  But 
it  cannot  compete  with  the  steam-engine  as  long  as  zinc  and  acids 
cost  so  much  more  than  coal. 

502.  Electro-magnetic  Telegraph.— To  our  countryman, 
Prof.  S.  F.  B.  Morse,  is  due  the  credit  of  the  erection  of  the  first 
telegraph  line  in  the  United  States.    It  extended  from  Baltimore 
to  Washington,  and  went  into  operation  in  1844. 

Communication  in  various  ways  by  means  of  electricity  be- 
tween places  a  few  miles  apart  was  not  unknown  in  Europe  before 
that  time,  and  several  ingenious  systems  have  appeared  since. 
One  of  these  is  Wheatstone's,  which  is  commonly  used  in  England. 
But  the  Morse  system  has  been  very  generally  preferred  on  ac- 
count of  its  greater  simplicity  and  efficiency,  and  it  is  now  widely 
used  in  the  United  States  and  on  the  continent  of  Europe,  where 
it  is  known  as  the  American  system.  The  principle  of  its  opera- 
tion is  as  follows : 

Let  E  of  Fig.  280  be  furnished  with  a  style  e  (Fig.  282)  directly 
over  which  is  the  groove  on  the  surface  of  a  solid  brass  roDer  c. 
Between  c  and  e  is  the  long  paper  ribbon  R  R.  Also  let  A  be 


304 


DYNAMICAL    ELECTRICITY. 


placed  above  M  and  be  furnished  with  a  spring  s  to  raise  it  as  far 
as  the  screw  i  allows  when  it  is  not  attracted  by  M.  When  the 
circuit  is  closed,  A  is  attracted  and  e  rises  and  forces  the  paper 


FIG.  282. 


into  the  groove,  producing  a  slight  elevation  on  its  upper  surface. 
The  ribbon  is  pulled  along  at  a  uniform  rate  in  the  direction  of 
the  arrow  by  clockwork  (not  shown  in  the  figure),  so  that  when 
the  circuit  remains  closed  for  a  little  time,  a  dash  is  marked  on 
the  paper  by  e ;  when  it  is  closed  and  instantly  opened,  the  result 
is  a  dot — or  rather  a  very  short  dash.  Spaces  are  left  between 
these  whenever  the  circuit  is  opened.  Combinations  of  these  dots, 
dashes,  and  spaces,  all  carefully  regulated  in  length,  compose  the 
letters  of  the  alphabet.  Spaces  are  also  left  between  the  letters, 
and  longer  ones  between  words. 

By  lengthening  the  circuit  wire,  it  is  evident  that  the  person 
who  sends  the  message  afc  n  p,  and  the  one  who  receives  it  at  E, 
may  be  miles  apart,  and  the  transmission  will  be  almost  instanta- 
neous owing  to  the  rapid  passage  of  the  current. 

The  essential  parts  of  this  system,  or  indeed  of  any  system,  are 
a  communicator  at  n  p,  an  indicator  at  E,  and  a  wire  extending 
from  one  to  the  other. 

503.  The  Connecting  Wire. — It  was  at  first  supposed  that 
a  complete  metallic  circuit  was  necessary,  hence  a  return  wire  was 

FIG.  283. 


employed.    But  this  was  rejected  when  it  was  found  that  the 
earth  could  be  used  as  a  part  of  the  circuit,  as  shown  in  Fig.  283. 


THE  INDICATOR.  305 

S'  are  the  terminal  stations,  and  s  is  one  of  the  way  stations 
which  may  occur  anywhere  along  the  line.  At  every  station  both 
a  communicator,  C,  and  indicator,  /,  are  introduced  into  the  cir- 
cuit, so  that  messages  can  be  both  sent  and  received. 

504.  The  Communicator. — This  consists  of  a  lever,  I  (Fig. 
284),  and  anvil,  a,  both  of  brass,  and  insulated  from  each  other. 

FIG.  284. 


The  anvil  connects  with  the  line  wire  W,  and  I  with  the  rest  of  the 
circuit  through  W,  and  W  W'  of  the  next  figure.  (See  also  Fig. 
283.)  The  end  of  I  is  depressed  by  the  finger  of  the  operator  on  the 
insulating  button  b,  and  is  raised  by  the  spring  s  when  the  pressure 
is  removed.  The  former  movement  closes  the  circuit,  the  latter 
opens  it,  and  by  a  succession  of  these  the  message  is  sent. 

When  the  communicator  is  not  in  use,  the  brass  bar  Jc  hinged 
to  the  base  of  I  is  pressed  into  contact  with  a.  This  closes  the 
circuit  for  oth£r  stations  on  the  line,  and  hence  Tc  is  called  the  cir- 
cuit closer.  The  whole  apparatus  is  called  the  key. 

505.  The  Indicator. — This  consists  of  two  parts,  (1st)  the 
relay,  and  (3d)  either  the  register,  or  the  sounder. 

The  first  part,  called  the  relay,  or  the  relay  magnet  (Fig.  285), 
consists  of  an  electro-magnet,  an  armature,  a  lever,  and  a  spring, 
the. same  as  in  Fig.  282,  except  that  the  electro-magnet  is  horizon- 
tal, and  the  other  parts  correspond  in  position.  The  tension  of 
the  spring  s  is  regulated  by  the  screw  and  milled  head  h,  and  M 
is  adjusted  by  a  similar  screw  (between  W  and  W"  in  the  figure), 
which  slides  it  along  the  grooved  way  X.  One  end  of  the  coil 
wire  passes  out  through  W  to  W  of  Fig.  284.  The  other  end 
connects  at  IF  "  with  one  pole  of  the  battery  if  it  is  at  S  (Fig.  283), 
with  the  earth  if  it  is  at  S',  or  with  the  line  wire  to  the  next  sta- 
tion if  it  is  at  s. 

The  reason  for  introducing  the  relay  is  this:  The  current 
from  the  preceding  station  has  become  too  feeble  to  cause  indenta- 
tion of  the  paper  by  the  style,  and  thus  make  a  visible  record,  or 
even  to  produce  a  distinct  sound  of  the  armature  upon  the  mag- 
net for  reading  messages  by  the  ear.  The  relay  is  therefore  con* 
trived  for  employing  this  feeble  current  to  close  and  open  the 
20 


306 


DYNAMICAL    ELECTRICITY. 


circuit  of  a  local  battery,  whose  current  is  powerful  enough  to  de- 
liver messages  in  either  form,  or  even  in  both  forms  at  once.  All 
which  the  weak  current  of  the  distant  battery  has  to  do  is  to  cause 
the  armature  A  to  move  toward  the  magnet  M  till  the  top  of  L 
touches  the  screw  N,  and  thus  closes  the  circuit  of  the  local  bat- 
tery. When  the  current  ceases,  a  delicate  spring,  s,  draws  L  back 
from  contact  with  N,  and  hreaks  the  circuit  of  the  local  hatterv. 


By  the  adjustments  ahove  described,  the  distance  through  which 
L  moves,  and  the  force  of  the  spring  s,  may  he  made  as  small  as 
the  operator  pleases. 

506.  The  Register  and  the  Sounder.— The  second  part  of 
the  indicator  is  either  a  register,  or  a  sounder,  according  as  mes- 
sages are  to  he  addressed  to  the  eye  or  to  the  ear.  The  register 
(Fig.  282)  has  been  already  described ;  the  current  of  the  local 
battery  close  at  hand  has  force  enough  to  cause  visible  iudenta- 


FIG. 


tions  in  the  paper  whenever  the  lever  is  drawn  to  the  magnet ; 
and  this' record  can  be  read  at  any  time  subsequently.  Within  a 
*few  years  a  modified  form  of  the  register  has  come  into  use,  and 
is  called  the  sounder.  In  this  the  end  of  the  lever  L'  (Fig.  286), 


REPEATERS.  307 

instead  of  being  furnished  with  a  style,  is  made  to  strike  against 
the  two  screws,  N',  0'.  The  downward  click  is  a  little  louder 
than  the  upward  one,  and  so  the  beginning  and  end  of  each  dot 
or  dash  are  distinguished  from  each  other.  Many  operators  learn 
from  the  first  to  read  ly  the  ear,  and  have  never  used  a  register. 

Whether  a  register  or  a  sounder  is  employed,  its  coil  wire  is 
entirely  distinct  from  the  line  wire,  and  belongs  only  to  the  local 
battery.  The  circuit  of  this  battery  may  be  traced  (Figs.  285,  286) 
from  the  positive  pole  through  I,  L,  N,  n,  o, . . . .  o,  n'9  the  coil  of 
the  sounder,  and  n"9  to  the  negative  pole.  The  binding  screws,  I, 
n,  n',  n",  are  connected  with  their  respective  levers,  or  contact 
screws,  by  insulated  wires  concealed  in  the  bases.  "When  A  is  at- 
tracted by  M,  L  touches  N9  and  the  circuit  is  closed ;  when  it  is 
withdrawn  by  s9  the  circuit  is  opened,  because  0  is  insulated. 
Hence  the  motions  of  L  and  L'  are  simultaneous. 

Since  the  relay  is  always  in  the  main  circuit,  it  communicates, 
by  means  of  the  local  current,  to  the  operator  at  whose  station  it 
is,  all  the  messages  sent  between  any  two  stations  on  the  line,  in- 
cluding those  which  he  himself  sends.  Hence,  if  his  own  indi- 
cator does  not  operate  while  he  is  at  work,  he  knows  that  his  mes- 
sage is  not  passing  over  the  line,  owing  to  some  break  in  the 
circuit. 

507.  Repeaters. — On  a  well  insulated  wire  the  weakness  of 
the  current  at  the  distance  of  a  few  miles  from  the  battery  is 
mainly  due  to  the  resistance  of  the  wire.  The  nature  of  this  re- 
sistance is  unknown,  but  it  is  subject  to  the  same  law  as  the  fric- 
tion of  a  fluid  along  the  interior  of  a  tube ;  it  varies  directly  as 
the  length,  and  inversely  as  the  diameter.  Hence  telegraph  wires 
of  considerable  thickness  are  employed,  and  even  then,  after  a  cer- 
tain number  of  miles,  varying  according  to  strength  of  battery, 
insulation,  etc.,  the  current  will  not  work  even  a  relay.  Before 
reaching  that  point,  therefore,  the  wire  is  allowed  to  pass  into  the 
ground,  and  so  complete  the  circuit. 

To  pass  a  message  beyond  the  place  at  which  the  current  will 
only  work  a  relay,  N'  and  L'  are  made  parts  of  a  new  circuit 
called  a  repeater,  which  is  closed  and  opened  simultaneously  with 
the  preceding  one  by  the  motions  of  L',  just  as  a  local  circuit  is 
worked  by  L  (Art.  505). 

On  the  28th  of  February,  18G8,  signals  were  sent  through 
from  Cambridge,  Mass.,  to  San  Francisco,  by  the  employment  of 
thirteen  repeaters.  The  time  occupied  by  the  signals  in  going 
and  returning  (making  about  7,000  miles)  was  three-tenths  of  a 
second — allowance  being  made  for  the  coil  wires  of  the  electro- 
magnets through  which  the  current  passed. 


308  DYNAMICAL    ELECTRICITY. 

5D8.  Atlantic  Telegraph  Cable. — This  cable  stretches  a 
distance  of  3,500  miles,  and  from  the  nature  of  the  case  is  a  con- 
tinuous wire,  so  that  it  cannot  be  advantageously  worked  by  the 
Morse  apparatus.  The  indicator  employed  is  a  sensitive  galva- 
nometer needle  which  is  made  to  oscillate  on  opposite  sides  of  the 
zero  point  by  the  passage  through  it  of  currents  in  opposite  direc- 
tions. But  to  reverse  the  direction  of  the  current  throughout  the 
whole  length  of  the  cable  is  a  slow  process.  For  the  cable  is  an 
immense  Ley  den  jar,  the  surface  of  the  copper  wire  (amounting  to 
425,000  sq.  feet)  answering  to  the  inner  coating,  the  water  of  the 
ocean  to  the  outer,  and  the  gutta-percha  between  the  two  to  the 
glass  of  an  ordinary  jar.  A  current  passing  into  it  is  therefore 
detained  by  electricity  of  the  contrary  kind  induced  in  the  water, 
and  no  effect  will  be  produced  at  the  further  end  until  it  is 
charged. 

This  very  circumstance,  at  first  considered  a  misfortune,  is  now 
taken  advantage  of  in  a  very  simple  and  ingenious  manner  to 
facilitate  the  transmission  of  signals.  The  current  is  allowed  to 
pass  into  the  cable  till  it  is  charged — then,  without  breaking  the 
circuit,  by  depressing  a  key  for  an  instant,  a  connection  is  made 
between  it  and  a  wire  running  out  into  the  sea;  that  is,  between 
the  inner  and  outer  coatings.  This  partially  discharges  it,  and 
the  needle  at  the  other  end  is  deflected.  "When  the  key  is  raised 
the  discharge  ceases,  the  current  flows  on  as  before,  and  the  needle 
is  deflected  in  the  opposite  direction. 

It  is  said  that  after  this  plan  was  adopted,  twenty  words  could 
be  sent  through  the  cable  per  minute,  whereas  only  four  per  min- 
ute could  be  sent  before.  The  greatest  speed  thus  far  attained  on 
land  wires  is  believed  to  have  been  the  transmission  in  one  in- 
stance of  1,352  words  in  thirty  minutes  between  New  York  and 
Philadelphia  in  1868. 

509.  Fire-Alarm  Telegraph. — Recurring  again  to  the  stand- 
ard (Fig.  280),  the  end  E  may  be  so  connected  with  machinery  as 
to  cause  the  striking  of  a  bell  in  a  distant  tower  whenever  the  cir- 
cuit is  closed  at  n  p.    In  our  large  cities  boxes  are  placed  at  con- 
venient points,  each  containing  a  crank,  or  lever,  by  which  the 
circuit  may  be  closed  and  the  fire-bell  rung.     Thus,  by  previously 
arranged  signals,  the  locality  of  the  fire  is  immediately  made 
known  at  the  various  engine-houses. 

510.  Chronograph. — This  is  used  in  observatories  for  record- 
ing the  passage  of  stars  across  the  meridian.    Imagine  the  circuit 
of  Fig.  282  to  be  closed  and  instantly  broken  again,  by  a  clock 
pendulum  at  the  end  of  every  second.    As  the  paper,  R  R,  moves 
uniformly,  dots  are  made  on  it  at  equal  distances  from  each  other, 


CHRONOGRAPH.  399 

each  of  which  distances,  therefore,  represents  one  second.  The 
observer  has  a  key,  by  which  also  he  closes  the  circuit  for  an  in- 
stant when  a  certain  star  passes  the  meridian.  The  dot  thus  made 
shows,  by  its  situation  between  the  two  nearest  second  dots,  at 
what  fraction  of  the  second  the  transit  occurred. 

In  practice,  however,  the  record  is  more  conveniently  made  on 
a  large  sheet  of  paper,  which  is  wrapped  tightly  around  a  cylinder. 
The  clock-work,  which  revolves  the  cylinder,  also  moves  the  re- 
cording pen  in  a  line  parallel  to  its  axis.  By  these  two  motions,  a 
spiral  ink-line  is  traced  on  the  paper.  At  the  end  of  every  beat 
of  the  observatory  clock,  the  closing  of  the  circuit  gives  the  pen  a 
momentary  lateral  movement,  by  which  a  slight  notch  is  made  in 
the  line.  A  similar  notch  is  made  by  the  touch  of  the  key,  when 
the  observer  perceives  the  star  on  the  meridian  wire  of  the  tele- 
scope. Fig.  287  represents  a  portion  of  the  sheet  after  its  removal 

FIG.  287. 


from  the  cylinder;  #,  5,  c,  d,  &c.,  are  the  second  marks;  x,  y,  z, 
&c.,  are  transit  records.  The  ratio  my.mn  shows  what  fraction 
of  the  second  m  n  has  elapsed  when  the  transit  y  occurs. 


PART    YIII 

E  .A.  T . 


CHAPTER    I. 

EXPANSION  BY  HEAT.— THE  THERMOMETER. 

511.  Nature  of  Heat. — Heat  is  another  of  those  agencies 
which  have  been  regarded  as  imponderable  substances.    It  was 
said  to  emanate  in  straight  lines  from  the  sun  and  from  bodies  in 
combustion,  to  cause  the  sensation  of  warmth  when  it  strikes  us, 
to  expand  bodies  when  it  enters  them,  to  raise  their  temperature, 
&c.    But  there  is  abundant  reason  for  believing  that  heat  consists 
of  exceedingly  minute  and  rapid  vibrations  of  ordinary  matter  and 
of  the  ether  which  fills  all  space.    It  is  to  be  regarded  as  one  of 
the  modes  of  motion,  which  may  be  caused  by  any  kind  of  force, 
and  which  may  be  made  a  measure  of  that  force.    Heat  affects 
only  one  of  our  senses,  that  of  feeling.     Its  increase  produces  the 
sensation  of  warmth,  and  its  diminution  that  of  cold. 

512.  Expansion  and  Contraction  by  Heat  and  Cold.— 

It  is  found  to  be  a  fact  almost  without  exception,  that  as  bodies 
are  heated  they  are  expanded,  and  that  they  contract  as  they  are 
cooled.  It  is  easy  to  conceive  that  the  vibratory  motion  of  the 
several  molecules  of  a  body  compels  them  to  recede  from  each 
other,  and  to  recede  the  more  as  the  vibration  becomes  more  vio- 
lent. Although  the  change  in  magnitude  is  generally  very  small, 
yet  it  is  rendered  visible  by  special  contrivances,  and*is  made  the 
means  of  measuring  temperature. 

513.  The   Thermometer. — This  instrument  measures  the 
degree  of  heat,  or  the  temperature,  of  the  medium  around  it,  by 
the  expansion  and  contraction  of  some  substance.     The  substance 
commonly  employed  is  mercury.    The  liquid,  being  inclosed  in  a 
glass  bulb,  can  expand  only  by  rising  in  the  fine  bore  of  the  stem, 
where  very  small  changes  of  volume  are  rendered  visible.     A 


EXPANSION.  311 

scale  is  attached  to  the  stem  for  reading  the  degrees  of  tempera- 
ture, 

The  graduation  of  the  thermometer  must  begin  with  the  fixing 
of  two  important  points  by  natural  phenomena,  the  freezing  and 
boiling  of  water.  When  the  bulb  is  plunged  into  powdered  ice, 
the  point  at  which  the  column  settles  is  the  freezing  point  of  the 
thermometer.  And  if  it  is  placed  in  pure  boiling  water  under  a 
given  atmospheric  pressure,  the  mercury  indicates  the  boiling  point. 
Between  these  two  points,  namely  32°  and  212°  R,  there  must  be 
180°,  and  the  scale  is  graduated  accordingly.  As  the  bore  of  the 
tube  is  not  likely  to  be  exactly  equal  in  all  parts,  the  length  of  the 
degrees  should  vary  inversely  as  the  area  of  the  cross-section. 
This  is  accomplished  by  moving  a  short  column  of  mercury  along 
the  different  parts  and  comparing  the  lengths  occupied  by  it.  The 
degrees  in  the  several  parts  must  vary  in  the  same  ratio. 

514.  Different  Systems  of  Graduation. — There  are  in  use 
three  kinds  of  thermometer  scale,  Fahrenheit's,  Reaumur's,  and 
the  Centigrade.     In  Fahrenheit's,  the  freezing  point  of  water  is 
called  32°,  and  the  boiling  point  212° ;  in  Reaumur's,  the  freezing 
point  is  called  0°,  and  the  boiling  point  80° ;  in  the  Centigrade, 
the  freezing  point  0°,  and  the  boiling  point  100°.     In  a  scientific 
point  of  view,  the  Centigrade  is  preferable  to  either  of  the  others, 
but  Fahrenheit's  is  generally  used  in  this  country.     The  letter  F., 
R.,  or  C.,  appended  to  a  number  of  degrees,  indicates  the  scale 
intended.    In  this  country,  F.  is  understood  if  no  letter  is  used. 

515.  To  Reduce  from  one   Scale  to  Another. — Since 
the  zero  of  Fahrenheit's  scale  is  32°  below  the  freezing  point,  while 
in  both  of  the  others  it  is  at  the  freezing  point,  32°  must  always 
be  subtracted  from  any  temperature  according  to  Fahrenheit,  in 
order  to  find  its  relation  to  the  zero  of  the  other  scales.    Then* 
since  212° -32°  (  =  180°)  F.  are  equal  to  80°  R.,and  to  100°  C.,  the 
formula  for  changing  F.  to  R.  is  |  (F.— 32)  =R. ;  and  for  changing 
F.  to  C.,  it  is  f  (F.-32)=C.    Hence,  to  change  R-  to  F.,  we  have 
|  R.  +  32=F.;  and  to  change  C.  to  F.,  f  C.  +  32=F. 

Mercury  congeals  at  about  —39°  F. ;  therefore,  for  tempera- 
tures lower  than  that,  alcohol  is  used,  which  does  not  congeal  at 
any  known  temperature. 

516.  Expansion  of  Solids.— When  the  expansion  of  a  solid 
is  considered  simply  in  one  dimension,  it  is  called  linear  expan- 
sion; in  two  dimensions  only,  superficial  expansion;  in  all  three 
dimensions,  cubical  expansion. 

The  linear  expansion  of  a  metallic  rod  is  readily  made  visible 
by  an  instrument  called  the  pyrometer,  which  magnifies  the  mo- 


312  HEAT. 

tion.  The  end  A  of  the  rod  A  B  (Fig.  288)  is  held  in  place  by  a 
screw.  The  end  B  rests  against  the  short  arm  of  the  lever  (7,  the 
longer  arm  of  which  bears  on  the  arm  D  of  the  long  bent  lever 

FIG.  288. 


D  E\  this  serves  as  an  index  to  the  graduated  arc  E  F.  The  long 
metallic  dish  G  G,  being  raised  on  the  hinges  H  Hy  so  as  to  en- 
close the  bar  A  B,  and  then  filled  with  hot  water,  the  bar  instantly 
expands,  and  raises  the  index  along  the  arc  E  F. 

517.  Coefficient  of  Expansion. — The  coefficient  of  linear 
expansion  of  a  given  substance  is  the  fractional  increase  of  its 
length,  when  its  temperature  is  raised  one  degree.    But  since  this 
increase  is  generally  somewhat  greater  at  higher  temperatures,  the 
coefficients  of  expansion  given  in  tables  usually  refer  to  a  temper- 
ature at  or  near  the  freezing  point  of  water.    Thus,  the  coefficient 
of  expansion  for  silver  is  0.00001061 ;  by  which  is  meant  that  a 
silver  bar  one  foot  long  at  32°  F.  becomes  1.00001061  ft.  in  length 
at  33°  F. 

The  coefficient  of  superficial  expansion  is  twice,  and  that  of 
cubical  expansion  three  times  as  great  as  the  coefficient  of  linear 
expansion.  For,  suppose  c  to  be  the  coefficient  of  linear  expan- 
sion ;  then  if  the  edge  of  a  cube  is  1,  and  the  temperature  is  raised 
1°,  the  edge  becomes  1+c,  and  the  area  of  one  side  becomes 
(l+cY= l+%c  +  c\  and  the  volume  (I+cY=l  +  3c  +  3c*-\-c3.  But 
as  c  is  very  small,  the  higher  powers  may  be  neglected,  and  the 
area  is  1  +  %c,  and  the  volume  is  1 4-  3c ;  that  is,  the  coefficient  of 
superficial  expansion  is  2c,  and  that  of  cubical  expansion  is  3c,  as 
stated  above. 

518.  The  Coefficient  of  Expansion  differs  in  different 
Substances. — Copper  expands  nearly  twice  as  fast  as  platinum; 
the  ratio  of  expansion  in  steel  and  brass  is  about  as  61  to  100. 
This  ratio  is  employed  in  the  construction  of  the  compensation 
pendulum  (Art.  171).     The  same  is  sometimes  used  also  to  render 
constant  the  length  of  the  rod  with  which  the  base  line  of  a  trigo- 
nometrical survey  is  measured. 


EXPANSION    OF    LIQUIDS.  313 

If  two  thin  slips  of  metal  of  different  expansibility  be  soldered 
together  so  as  to  make  a  slip  of  double  thickness,  it  will  bend  one 
way  and  the  other  by  changes  of  temperature.  If  it  is  straight 
at  a  certain  temperature,  heating  will  bend  it  so  as  to  bring  the 
most  expansible  metal  on  the  convex  side  ;  and  cooling  will  bend 
it  in  the  opposite  direction  ;  and  the  degree  of  flexure  will  be  ac- 
cording to  the  degree  of  change  in  temperature.  Compensation 
in  clocks  and  watches  is  sometimes  effected  on  this  plan.  If  the 
compound  slip  has  the  form  of  a  helix,  with  the  most  expansible 
metal  on  the  inside,  heating  will  begin  to  uncoil  it,  and  cooling, 
to  coil  it  closer.  A  very  sensitive  thermometer,  known  as  Bre- 
guet's  thermometer,  is  constructed  on  this  principle. 

519.  The  Strength  of  the  Thermal  Force.— It  is  found 
that  the  force  exerted  by  a  body,  when  expanding  by  heat  or  con- 
tracting by  cold,  is  equal  to  the  mechanical  force  necessary  to  ex- 
pand or  compress  the  body  to  the  same  degree.  The  force  is  there- 
fore very  great.  If  the  rails  were  to  be  fitted  tightly  end  to  end 
on  a  railroad,  they  would  be  forced  out  of  their  places  by  expan- 
sion in  warm  weather,  and  the  track  ruined.  The  tire  of  a  car- 
riage wheel  is  heated  till  it  is  too  large,  and  then  put  upon  the 
wheel ;  when  cool,  it  draws  together  the  several  parts  with  great 
firmness.  In  repeated  instances,  the  walls  of  a  building,  when 
they  have  begun  to  spread  by  the  lateral  pressure  of  an  arched 
roof,  have  been  drawn  together  by  the  force  of  contraction  in  cool- 
ing. A  series  of  iron  rods  being  passed  across  the  building  through 
the  upper  part  of  the  walls,  and  broad  nuts  being  screwed  upon 
the  ends,  the  alternate  bars  are  expanded  by  the  heat  of  lamps, 
and  the  nuts  tightened.  Then,  when  they  cool,  they  draw  the 
walls  toward  each  other.  The  remaining  bars  are  then  treated  in 
the  same  manner,  and  the  process  is  repeated  till  the  walls  are 
restored  to  their  vertical  position  and  secured.  For  a  measure  of 
the  force  of  heat  see  Art.  555. 

520. — Expansion  of  Liquids. — It  lias  already  been  noticed 
that  mercury  and  alcohol  expand  by  heat,  and  are  therefore  used 
in  thermometers  for  measuring  temperature.  These  are  the  best 
liquids  for  such  a  purpose,  because  their  temperature  of  congela- 
tion is  very  low. 

As  liquids  have  no  permanent  form,  the  coefficient  of  expan- 
sion for  them  is  always  understood  to  be  that  of  cubical  expansion. 
There  is  a  practical  difficulty  in  the  way  of  finding  the  coefficient 
for  liquids,  because  they  must  be  enclosed  in  some  solid,  which 
also  expands  by  heat.  Hence,  the  apparent  expansion  must  be 
corrected  by  allowing  for  the  expansion  of  the  inclosing  solid,  be- 
fore the  coefficient  of  absolute  expansion  is  known. 


314  HEAT. 

This  fact  is  illustrated  by  the  following  experiment.  Fill  the 
bulb  and  part  of  the  stem  of  a  large  thermometer  tube  with  a  col- 
ored liquid,  and  then  plunge  the  bulb  quickly  into  hot  water ;  the 
first  effect  is,  that  the  liquid  falls,  as  if  it  were  cooled  ;  after  a  mo- 
ment it  begins  to  rise,  and  continues  to  do  so  till  it  attains  the 
temperature  of  the  hot  water.  The  first  movement  is  caused  by 
the  expansion  of  the  glass,  which  is  heated  so  as  to  enlarge  its 
capacity  and  let  down  the  liquid  before  the  heat  has  penetrated 
the  latter.  It  is  obvious  that  what  is  rendered  visible  in  this  case, 
must  always  be  true  when  a  liquid  is  heated — namely,  that  the 
vessel  itself  is  enlarged,  and  therefore  that  the  rise  of  the  liquid 
shows  only  the  difference  of  the  two  expansions.  Ingenious 
methods  have  been  devised  for  obtaining  the  coefficients  of  abso- 
lute expansion  of  liquids,  and  the  results  are  to  be  found  in  tables 
on  this  subject. 

521.  Exceptional  Case. — There  is  a  very  important  excep- 
tion to  the  general  law  of  expansion  by  heat  and  contraction  by 
cold,  in  the  case  of  water  just  above  the  freezing  point.    If  water 
be  cooled  down  from  its  boiling  point,  it  continually  contracts  till 
it  reaches  a  point  somewhat  above  39°  F.,  when  it  begins  to  ex- 
pand, and  continues  to  expand  till  it  freezes  at  32°  F.     On  the 
other  hand,  if  water  at  32°  F.  be  heated,  it  contracts  till  it  readies 
a  point  between  39°  and  40°  F.,  when  it  commences  to  expand. 
Therefore  the  density  of  water  is  greatest  at  the  point  where  this 
change  occurs.    Different  experimenters  vary  a  little  as  to  its  ex- 
act place,  but  it  is  usually  called  4°  C.,  or  39.2°  F. 

The  importance  of  this  exception  is  seen  in  the  fact  that  ice 
forms  on  the  surface  of  water,  and  continues  to  float  until  it  is 
again  dissolved.  As  the  cold  of  winter  comes  on,  the  upper  stra- 
tum of  a  lake  grows  more  dense  and  sinks ;  and  this  process  con* 
tinues  till  the  temperature  of  the  surface  reaches  39°,  when  it  is 
arrested.  Below  that  point  the  surface  grows  lighter  as  it  becomes 
colder,  till  ice  is  formed,  and  shields  the  water  beneath  from  the 
severe  cold  of  the  air  above. 

As  in  solids  so  in  liquids,  the  thermal  force  is  very  great.  Sup- 
pose mercury  to  be  expanded  by  raising  its  temperature  one  de- 
gree, it  would  require  more  than  300  pounds  to  the  square  inch  to 
compress  it  to  its  former  volume. 

522.  Expansion  of  Gases.— The  gases  expand  by  heat  more 
rapidly  and  more  regularly  than  solids  and  liquids.     The  large  ex- 
pansion and  contraction  of  air  is  made  visible  by  immersing  the 
open  end  of  a  large  thermometer  tube  in  colored  liquid.    When  the 
bulb  is  warmed,  bubbles  of  air  are  forced  out  and  rise  to  the  top 


RADIATION.  315 

of  the  liquid ;  when  it  is  cooled,  the  air  contracts  and  the  liquid 
rises  rapidly  in  the  tube. 

The  coefficient  of  expansion  for  air  is  about  0.00205,  which  in- 
creases slightly  with  increase  of  temperature  and  of  pressure. 
And*  most  of  the  gases  have  coefficients  which  differ  but  little 
from  this. 


CHAPTER    II. 

PASSAGE  OF  HEAT  THROUGH  SPACE  AND  MATTER. 

523.  Heat  is  Communicated  in  Several  Ways. — 1.  By 

radiation.  Heat  is  said  to  be  radiated  when  the  vibratory  motion 
is  transmitted  from  the  source  with  great  swiftness  through  the 
ether  which  fills  space.  Its  velocity  is  supposed  to  be  the  same  as 
that  of  light.  The  motion  is  propagated  in  straight  lines  in  every 
direction,  and  each  line  is  called  a  ray  of  heat.  We  feel  the  rays 
of  heat  from  the  sun  or  a  fire,  when  no  object  intervenes  between 
it  and  ourselves. 

2.  By  reflection.    When  rays  of  heat,  on  striking  a  surface,  are 
thrown  back  from  it,  they  are  said  to  be  reflected;  and  the  law  of 
reflection  is  the  same  as  for  sound,  namely,  the  angle  of  incidence 
equals  the  angle  of  reflection,  and  they  lie  on  opposite  sides  of  the 
perpendicular  to  the  surface. 

3.  By  conduction.    This  is  the  slow  progress  of  the  vibratory 
motion  from  one  atom  to  another  of  ordinary  matter. 

4.  By  convection.^    This  mode  of  communication  takes  place 
only,  in  fluids.    When  the  particles  are  expanded  by  heat,  they  are 
pressed  upward  by  others  which  are  colder  and  therefore  specifi- 
cally heavier.    Heat  is  thus  conveyed  from  place  to  place  by  the 
motion  of  the  heated  matter. 

524.  Radiation  of  Heat. — The  intensity  of  heat  radiated 
from  a  given  kind  of  source,  is  governed  by  the  three  following 
laws : 

1.  The  intensity  of  radiated  heat  varies  as  the  temperature  of 
the  source. 

2.  It  varies  inversely  as  the  square  of  the  distance. 

3.  It  groivs  less,  while  the  inclination  of  the  rays  to  the  surface 
of  the  radiant  groivs  less. 

The  truth  of  these  laws  is  ascertained  by  a  series  of  careful  ex- 
periments.   But  the  second  may  be  proved  mathematically  from 


316  HEAT. 

the  fact  of  propagation  in  straight  lines,  and  is  true  of  other  ema- 
nations, such  as  sound  and  light.  For  the  heat,  as  it  advances  in 
every  direction  from  the  radiant,  is  spread  over  spherical  surfaces 
which  increase  as  the  squares  of  the  distances ;  therefore  the  in- 
tensities must  grow  less  in  the  same  ratio ;  that  is,  the  intensities 
vary  inversely  as  the  squares  of  the  distances. 

TJie  radiating  power  of  a  given  body  depends  on  the  condition 
of  its  surface. 

If  a  cubical  vessel  filled  with  hot  water  have  one  of  its  vertical 
sides  coated  with  lamp  black,  another  with  mica,  a  third  with  tar- 
nished lead,  and  the  fourth  with  polished  silver,  and  the  heat  ra- 
diated from  these  several  sides  be  concentrated  upon  a  thermome- 
ter bulb,  the  ratio  of  radiation  will  be  found  nearly  as  follows 

Lamp  black, 100 

Mica, 80 

Tarnished  lead, 45 

Polished  silver, 12 

Polished  metals  generally  radiate  feebly;  and  this  explains  the 
familiar  fact  that  hot  liquids  retain  their  temperature  much  better 
in  bright  metallic  vessels  than  in  dark  or  tarnished  ones. 

525.  Equalization  of  Temperature. — Eadiation  is  going 
on  continually  from  all  bodies,  more  rapidly  in  general  from  those 
most  heated ;  and  therefore  there  is  a  constant  tendency  toward 
an  equal  temperature  in  all  bodies.    A  system  of  exchange  goes 
on,  by  which  the  hotter  bodies  grow  cool,  and  the  colder  ones  grow 
warm,  till  the  temperature  of  all  is  the  same.    But  this  equality 
does  not  check  the  radiation ;  it  still  goes  forward,  each  body  im- 
parting to  others  as  much  heat  as  it  receives  from  them. 

526.  Reflection  of  Heat. — When  rays  of  heat  meet  the  sur- 
face of  a  body,  some  of  them  are  reflected)  passing  oif  at  the  same 
angle  with  the  perpendicular  on  the  opposite  side.    But  others 
pass  into  the  body,  and  are  said  to  be  absorbed  by  it.    It  is  true 
of  waves  of  heat  as  of  all  other  kinds  of  vibration,  that  when  they 
meet  a  new  surface  and  are  reflected,  the  angle  of  incidence  equals 
the  angle  of  reflection,  and  that  their  intensity  after  reflection  is 
weakened. 

If  a  person,  when  near  a  fire,  holds  a  sheet  of  bright  tin  so  as 
to  see  the  light  of  the  fire  reflected  by  it,  he  will  plainly  perceive 
that  heat  is  reflected  also.  And  if  any  sound  is  produced  by  the 
fire,  as  the  crackling  of  combustion,  or  the  hissing  of  steam  from 
wood,  the  reflection  of  the  sound  is  likewise  heard.  This  simple 
experiment  proves  that  waves  of  sound,  of  heat,  and  of  light,  all 
follow  the  same  law  of  reflection. 


ABSORPTION    OF    HEAT. 


317 


527.  Heat  Concentrated  by  Reflection.— Let  two  pol- 
ished reflectors,  M  and  N  (Fig.  289),  having  the  form  of  concave 
paraboloids,  be  placed  ten  or  fifteen  feet  apart,  with  their  axes  in 
the  same  straight  line,  and  let  a  red-hot  iron  ball  be  in  the  focus  A 


FIG.  289. 


of  one,  and  an  inflammable  substance,  as  phosphorus,  in  the  focus 
B  of  the  other ;  then  the  latter  will  be  set  on  fire  by  the  heat  of 
the  ball.  The  rays  diverging  from  A  to  J/are  reflected  in  parallel 
lines  to  N,  and  then  converged  to  B. 

If,  instead  of  phosphorus,  the  bulb  of  a  thermometer  is  put  in 
the  focus  B,  a  high  temperature  is  of  course  indicated  on  the  scale. 
Now  remove  the  hot  ball  from  A,  and  put  in  its  place  a  lump  of 
ice;  then  the  thermometer  at  B  sinks  far  below  the  temperature 
of  the  room.  This  last  experiment  does  not  prove  that  cold  is  re- 
flected as  well  as  heat,  but  confirms  what  was  stated  (Art.  525), 
that  all  objects  radiate  to  one  another  till  their  temperatures  are 
equalized.  The  ice  radiates  only  a  little  heat,  which  is  reflected  to 
the  thermometer,  but  the  latter  radiates  much  more,  which  is  re- 
flected to  the  ice,  so  that  the  temperature  of  the  thermometer 
rapidly  sinks. 

528.  Absorption  of  Heat. — So  much  of  the  radiant  heat  as 
falls  on  a  body  and  is  not  reflected,  is  absorbed.  The  absorbing 
power  in  a  body  is  found  to  be  in  general  equal  to  its  radiating 
power.  It  is  very  noticeable  that  bodies  equally  exposed  to  the 
radiant  heat  of  the  sun  or  a  fire,  become  very  unequally  heated. 
A  white  cloth  on  the  snow,  under  the  sunshine,  remains  at  the 


318  HEAT. 

surface ;  a  black  cloth  sinks,  because  it  absorbs  heat,  and  melts  the 
snow  beneath  it.  Polished  brass  before  a  fire  remains  cold ;  dark, 
unpolished  iron,  is  soon  hot. 

529.  Conduction  of  Heat  by  Solids.— While  radiated  and 
reflected  heat  moves  through  the  empty  spaces  of  the  solar  system, 
and  through  the  atmospheres  of  the  planets,  with  inconceivable 
velocity,  conducted  heat,  on  the  contrary,  passes  through  bodies 
very  slowly,  and  yet  at  very  different  rates  in  different  bodies. 
Those  in  which  heat  is  conducted  most  rapidly,  are  called  good 
conductors,  as  the  common  metals;  those  in  which  it  passes  slowly, 
are  called  poor  conductors,  as  glass  and  wood.    In  general,  the 
bodies  which  are  good  conductors  of  heat,  are  also  good  conduc- 
tors of  electricity.    Let  rods  of  different  metals  and  other  sub- 
stances, A,  B,  C,  &c.  (Fig.  290),  all  of  the  same  length,  be  inserted 
with  water-tight  joints  in  the 

side  of  a  wooden  vessel.    Then  FIG.  290. 

attach  by  wax  a  marble  under 
the  end  of  each  rod,  and  fill 
the  vessel  with  boiling  water. 
The  marbles  will  fall  by  the 
melting  of  the  wax,  not  at  the 
same,  but  at  different  times, 
showing  that  the  heat  reaches 
some  of  them  sooner  than  oth- 
ers. It  will  be  seen,  however,  in  the  chapter  on  specific  heat,  that 
the  order  in  which  they  fall  is  not  necessarily  the  order  of  con- 
ducting power. 

530.  Effects  of  Molecular  Arrangement— Organic  sub- 
stances usually  conduct  heat  poorly;  and  bodies  having  a  struc- 
tural arrangement  which  differs  in  different  directions,  are  not 
likely  to  conduct  equally  well  in  all  directions.    Thus,  let  two  thin 
plates  be  cut  from  the  same  crystal,  one,  A  (Fig.  291),  perpendic- 

FIG.  291. 


ular,  and  the  other,  B,  parallel  to  the  optic  axis.  Let  a  hole  be 
drilled  through  the  centre  of  each,  and  after  a  lamina  of  wax  has 
been  spread  over  the  crystal,  let  a  hot  wire  be  inserted  in  it.  On 


CONDUCTIVE    POWER.  319 

the  plate  A,  the  melting  of  the  wax  will  advance  in  a  circle,  show- 
ing equal  conducting  power  in  all  directions  in  the  transverse  sec- 
tion. In  the  plate  B,  it  will  advance  in  an  elliptical  form,  the 
major  axis  being  parallel  to  the  optic  axis  of  the  crystal,  proving 
the  best  conduction  to  be  in  that  direction. 

A  block  of  wood  cut  from  one  side  of  the  trunk  of  a  tree,  con- 
ducts most  perfectly  in  the  direction  of  the  fiber,  and  least  in  a 
direction  which  is  tangent  to  the  annual  rings  and  perpendicular 
to  the  fiber,  and  in  an  intermediate  degree  in  the  direction  of  the 
radius  of  the  rings. 

531.  Conduction  by  Fluids. — Fluids,  both  liquid  and  gas- 
eous, are  in  general  very  poor  conductors.    Water,  for  example, 
can  be  made  to  boil  at  the  top  of  a  vessel,  while  a  cake  of  ice  is 
fastened  within  it  a  few  inches  below  the   surface.      If  ther- 
mometers are  placed  at  different  depths,  while  the  water  boils  at 
the  top,  there   is  discovered  to  be   a  very  slight  conduction  of 
heat  downward.     The  gases  conduct  even  more  imperfectly  than 
liquids. 

It  will  be  seen  hereafter  (Art.  533)  that  a  mass  of  fluid  becomes 
heated  by  convection,  not  by  conduction. 

532.  Illustrations  of  Difference  in  Conductive  Power. — . 

In  a  room  where  all  articles  are  of  equal  temperature,  some  feel 
much  colder  than  others,  simply  because  they  conduct  the  heat 
from  the  hand  more  rapidly ;  painted  wood  feels  colder  than 
woolen  cloth,  and  marble  colder  still.  If  the  temperature  were 
higher  than  that  of  the  blood,  then  the  marble  would  seem  the 
hottest,  and  the  cloth  the  coolest,  because  of  the  same  difference 
of  conduction  to  the  hand. 

Our  clothing  does  not  impart  warmth  to  us,  but  by  its  non-con- 
ducting property,  prevents  the  vital  warmth  from  being  wasted  by 
radiation  or  conduction.  If  the  air  were  hotter  than  our  blood, 
the  same  clothing  would  serve  to  keep  us  cool. 

A  pitcher  of  water  can  be  kept  .cool  much  longer  in  a  hot  day, 
if  wrapped  in  a  few  thicknesses  of  cloth ;  for  these  prevent  the 
heat  of  the  air  from  being  conducted  to  the  water.  In  the  same 
way  ice  may  be  prevented  from  melting  rapidly. 

The  vibrations  of  heat,  like  those  of  sound,  are  greatly  inter- 
rupted in  their  progress  by  want  of  continuity  in  the  material. 
Any  substance  is  rendered  a  much  poorer  conductor  by  being  in 
the  condition  of  a  powder  or  fiber.  Ashes,  sand,  sawdust,  wool, 
fur,  hair.  &c.,  owe  much  of  their  non-conducting  quality  to  the 
innumerable  surfaces  which  heat  must  meet  with  in  being  trans- 
mitted through  them. 


330 


HEAT. 


FIG.  292. 


533.  Convection  of  Heat.— Liquids  and  gases  fire  heated 
almost  entirely  by  convection.    As  heat  is  applied  to  the  sides  and 
bottom  of  a  vessel  of  water,  the  heated  particles  become  specifically 
lighter,  and  are  crowded  up  by  heavier  ones  which  take  their  place. 
There  is  thus  a  constant  circulation  going  on  which  tends  to 
equalize  the  temperature  of  the  whole.     This  motion  is  made  visi- 
ble in  a  glass  vessel,  by  putting  into  the 

water  some  opaque  powder  of  nearly  the 
same  density  as  water.  Ascending  cur- 
rents are  seen  over  the  part  most  heated, 
and  descending  currents  in  the  parts  far- 
thest from  the  heat,  as  represented  in  Fig. 
292.  The  ocean  has  perpetual  currents 
caused  in  a  similar  manner.  The  hottest 
portions  flow  away  from  the  tropical  to- 
ward the  polar  latitudes,  while  at  greater 
depths  the  cold  waters  of  high  latitudes 
flow  back  towards  the  tropics. 

For  a  like  reason,  the  air  is  constantly 
in  motion.  The  atmospheric  currents 
on  the  earth  have  been  considered  in  Chap- 
ter III  of  Pneumatics. 

534.  Diathermancy.— It  has  already  been  noticed,  that  ra- 
diant heat  passes  freely  through  the  atmosphere  as  well  as  through 
vacant  space.    The  air  is  therefore  said  to  be  diathermal ;  it  is 
also  transparent,  since  it  permits  light  to  pass  freely  through  it. 
But  there  are  substances  which  allow  the  free  transmission  of  the 
waves  of  light,  but  not  those  of  heat;  and  there  are  others  through 
which  waves  of  heat  can  freely  pass,  but  not  those  of  light. 

Water  and  glass,  which  are  almost  perfectly  transparent  to  the 
faintest  light,  will  not  transmit  the  vibrations  of  heat  unless  they 
are  very  intense.  If  an  open  lamp-flame  shines  upon  a  thin  film 
of  ice,  while  nearly  the  whole  of  the  light  is  transmitted,  only  6 
per  cent,  of  the  heat  can  pass  through.  On  the  other  hand,  rock 
salt  is  remarkably  diathermal.  A  plate  of  it,  one-tenth  of  an  inch 
thick,  will  transmit  92  per  cent,  of  the  heat  of  a  lamp ;  and  if  if 
be  coated  with  lampblack  so  thick  as  to  stop  light  completely,  the 
heat  is  still  transmitted  with  almost  no  diminution. 

If  a  prism  be  made  of  a  substance  highly  diathermal,  as  rock 
salt,  it  is  found  that  heat,  as  well  as  light,  is  refracted,  being  bent 
from  its  course  less  than  most  of  the  colors,  arid  falling  mostly  be- 
yond the  red  extremity  of  the  visible  spectrum,  but  partially  coin- 
ciding with  that  color. 


SPECIFIC    HEAT.  321 

CHAPTER    III. 

SPECIFIC  HEAT.— CHANGES  OF  CONDITION.— LATENT  HEAT. 

535.  Specific  Heat. — The  heat  which  is  absorbed  by  a  body 
is  not  wholly  employed  in  raising  its  temperature.  While  a  part 
of  the  thermal  force  which  is  communicated,  throws  the  atoms 
into  vibration,  that  is,  heats  the  body,  another  part  performs  inte- 
rior work  of  some  other  kind,  such  as  urging  the  atoms  asunder, 
or  forcing  them  into  new  arrangements.  This  latter  portion  is 
lost  to  our  sense  and  to  the  thermometer,  until  the  body  is  again 
cooled,  when  it  re-appears.  The  relative  quantity  of  the  force  thus 
hidden  from  view  is  different  in  different  substances.  Hence  the 
phrase,  specific  heat,  is  used  to  express  the  amount  of  heat  required 
to  raise  a  given  weight  one  degree  of  temperature.  The  specific 
heat  of  water  is  greater  than  that  of  any  other  substance  known, 
and  it  is  made  the  standard  of  comparison. 

The  thermal  unit  is  the  amount  of  heat  required  to  raise  the 
temperature  of  a  pound  of  water  one  degree,  and  is  called  1.  The 
specific  heat  of  a  few  substances  is  given  in  the  following  table,  in 
order  to  show  how  greatly  they  differ. 


Water i.oooo 

Sulphur 0.2026 


Silver 0.0570 

Mercury    ....  0.0333 


Iron 0.1138      Gold 0.0324 

Copper  .     .     .     .     .  0.0951   I  Lead 0-0314 

If  a  pound  of  water,  a  pound  of  iron,  and  a  pound  of  mercury 
are  each  raised  one  degree  in  temperature,  the  water  consumes 
about  nine  times  as  much  heat  as  the  iron,  and  thirty  times  as 
much  as  the  mercury. 

When  bodies  are  cooled,  they  show  the  same  differences  in  the 
quantity  of  heat  which  they  give  off. 

It  is  a  benevolent  provision  in  nature,  that  water,  which  is  ex- 
tended over  so  large  a  portion  of  the  globe,  has  so  great  specific 
heat ;  for  the  changes  of  both  heat  and  cold  are  by  this  means 
greatly  moderated. 

536.  Method  of  Finding  Specific  Heat.— The  following 
is  one  of  .several  methods  of  finding  the  specific  heat  of  a  sub- 
stance ;  it  is  called  the  method  of  mixtures.  Let  a  known  weight 
of  the  substance  be  heated  to  a  certain  temperature,  and  then 
plunged  into,  or  mixed  with,  the  same  weight  of  water  of  a  low 
temperature ;  after  which  measure  the  temperature  of  the  mass. 
It  will  thus  be  known  how  much  one  has  lost  and  the  other 
21 


322  HEAT. 

gained  in  ordor  to  reach  the  common  point.  If  a  pound  of  mer- 
cury at  the  temperature  of  132°,  be  poured  into  a  pound  of  water 
at  the  temperature  of  32°,  the  mass  will  be  found  at  about  35.25°, 
the  water  being  heated  only  3.25°,  while  the  mercury  is  cooled 
96.75°.  Therefore  96.75° :  3.25°  : :  1  :  0.0335,  which  is  about  the 
specific  heat  of  mercury. 

The  specific  heat  of  bodies  is  in  general  a  little  greater  as  their 
temperature  rises.  That  of  the  gases,  however,  seems  to  be  nearly 
constant  at  all  temperatures,  and  under  all  pressures. 

537.  Apparent  Conduction  Affected  by  Specific  Heat. — 

The  conducting  power  of  different  substances  cannot  be  correctly 
compared,  without  making  allowance  for  their  specific  heat  (Art. 
529).  For  the  heat  which  is  communicated  to  one  end  of  a  rod, 
will  reach  the  other  end  more  slowly,  if  a  great  share  of  it  disap- 
pears on  the  way.  For  instance,  at  the  same  distance  from  the 
source  of  heat,  wax  is  melted  quicker  on  a  rod  of  bismuth  than 
on  one  of  iron,  though  iron  is  the  best  conductor,  because  the 
specific  heat  of  iron  is  three  times  as  great  as  that  of  bismuth  ; 
the  heat  actually  reaches  the- wax  soonest  through  the  iron,  but 
not  enough  to  melt  it,  because  so  much  is  required  to  raise  the 
iron  to  a  given  temperature. 

538.  Changes  of  Condition. — Among  the  most  important 
effects  produced  by  heat,  are  the  changes  of  condition  from  solid 
to  liquid  and  from  liquid  to  gas,  or  the  reverse,  according  as  the 
temperature  of  a  body  is  raised  or  lowered.    Increase   of  heat" 
changes  ice  to  water,  and  water  to  steam,  and  the  diminution  of 
heat  reverses  these  effects.    A  large  part  of  the  simple  substances, 
and  of  compound  ones  not  decomposed  by  heat,  undergo  similar 
changes  at  some  temperature  or  other ;  and  probably  it  would  be 
found  true  of  all  if  the  requisite  temperature  could  be  reached. 

The  melting  point  (called  also  freezing  point,  or  point  of  conge- 
latiori)  ol  a  substance  is  the  temperature  at  which  it  changes  from 
a  solid  to  a  liquid  or  the  reverse. 

The  loiling  point  is  the  temperature  at  which  it  changes  from 
a  liquid  to  a  gas  or  the  reverse. 

539.  Latent  Heat. — Whenever  a  solid  becomes  a  liquid,  or  a 
liquid  becomes  a  gas,  a  large  amount  of  heat  disappears,  and  is 
said  to  become  latent.     The  thermal  force  is  expended  in  sunder- 
ing the  atoms,  and  perhaps  in  putting  them  into  new  relations 
and  combinations,  so  that  there  is  not  the  slightest  increase  of 
temperature  after  the  change  begins  till  it  ends.     The  force  is  not 
lost,  but  is  treasured  up  in  the  form  of  potential  energy,  which  be- 


BOILING    UNDER    PRESSURE.  323 

comes  available  whenever  a  change  is  made  in  the  opposite  direc- 
tion. Using  the  force  of  heat  to  turn  water  into  steam,  is  like 
using  the  strength  of  the  arm  in  coiling  up  a  spring,  or  lifting  a 
weight  from  the  earth.  The  spring  and  the  weight  are  each  in  a 
condition  to  perform  work.  They  have  potential  energy,  which 
can  be  used  at  pleasure. 

It  has  been  already  noticed  that  much  heat  disappears  in 
bodies  of  great'  specific  heat,  as  their  temperature  rises.  But  the 
amount  which  becomes  latent,  while  a  change  of  condition  takes 
place,  is  vastly  greater.  Let  heat  be  applied  at  a  uniform  rate  to 
a  mass  of  water  at  the  temperature  of  32°,  until  it  rises  to  the 
boiling  point,  212°,  and  note  the  time  occupied.  Continuing  the 
same  uniform  supply,  it  will  require  5|  times  as  long  to  change  it 
all  into  steam.  In  other  words,  180°  of  heat  will  raise  water  from 
the  freezing  to  the  boiling  point,  and  (180°  x  5|  =)  967^°  are  re- 
quired to  change  the  same  into  steam,  which  still  remains  at  the 
temperature  of  212°  ;  the  whole  of  the  967°  of  thermal  force  have 
been  consumed  in  the  internal  work  of  re-arranging  the  atoms. 

540.  Temperature   of  Change  of  Condition. — Different 
substances  change  their  condition  at  very  different  temperatures. 
Water  solidifies  at  32°  R,  mercury  at  —  39°,  tin  at  455°,  gold  at 
201G°.    Water  boils  at  212°  F.,  ether  at  95°,  alcohol  at  173°,  mer- 
cury at  662°.    There  are  some  solids,  which  soften  gradually,  and 
pass  through  a  large  range  of  temperature  before  becoming  liquid, 
as  iron  and  glass.    No  definite  melting  point  can  be  given  for  such 
substances. 

Again,  many  liquids  pass  into  the  gaseous  state  by  a  slow  and 
almost  insensible  process  which  goes  on  at  the  surface.  This  is 
called  evaporation;  and  it  takes  place  at  all  temperatures,  but 
more  rapidly  as  the  temperature  is  higher.  Even  solids  evaporate 
without  passing  through  the  liquid  form.  Eor  example,  a  thin 
film  of  ice  on  a  pavement  wastes  away  in  cold  weather  without 
melting. 

541.  Boiling  under  Pressure. — The  boiling  point  for  water 
is  given  as  212°  F.    This  means  that  water  boils  at  that  point 
under  the  ordinary  pressure  of  the  air,  and  at  or  near  the  sea  level. 
At  that  temperature  the  steam  formed  has  a  tension  or  expansive 
force  equal  to  the  atmospheric  pressure.    But  if  the  pressure  were 
diminished,  water  would  boil  at  a  lower  temperature.     On  high 
mountains,  boiling  water  is  from  20°  to  30°  lower  in  temperature 
than  at  the  ocean  level.    And  under  the  receiver  of  an  air  pump, 
as  pressure  is  gradually  taken  off,  water  boils  at  lower  and  lower 
points  of  temperature  down  to  72°. 


324  HEAT. 

The  effect  of  diminished  pressure  to  lower  the  boiling  point  is 
well  shown  by  the  following  familiar  experiment :  In  a  thin  glass 
flask,  boil  a  little  water,  and  after  removing  it  from  the  lire,  cork 
and  invert  the  flask.  The  steam  which  is  formed  will  soon  press 
so  strongly  upon  the  water  as  to  stop  the  boiling.  When  this 
happens,  pour  a  little  cold  water  upon  the  flask ;  the  water  within 
will  immediately  commence  boiling  violently,  because  the  vapor  is 
condensed  and  the  pressure  removed.  This  effect  may  be  repro- 
duced several  times  before  the  water  in  the  flask  is  too  cool  to  boil 
in  a  vacuum. 

542.  Freezing    Produced    by   Melting.— Since  a    great 
amount  of  heat  disappears  in  a  substance  as  it  passes  from  the 
solid  to  the  liquid  state,  the  loss  thus  occasioned  may  produce 
freezing  in  a  contiguous  body.    When  salt  and  powdered  ice  are 
mixed,  their  union  causes  liquefaction.    And  if  this  mixture  is 
surrounded  by  bad  conductors,  and  a  tin  vessel  containing  some 
liquid  be  placed  in  the  midst  of  it,  the  latter  is  frozen  by  the  ab- 
straction of  heat  from  it,  by  the  melting  of  the  ice  and  salt.     In 
this  way  ice  creams  and  similar  luxuries  are  easily  prepared  in  hot 
as  well  as  in  cold  weather. 

543.  Freezing  by  Evaporation. — In  like  manner,  freezing 
by  evaporation  is  explained.     Put  a  little  water  in  a  shallow  dish 
of  thin  glass,  and  set  it  on  a  slender  wire-support  under  the  re- 
ceiver of  an  air  pump.     Beneath  the  wire-support  place  a  broad 
dish  containing  sulphuric  acid.    When  the  air  is  exhausted,  the 
water  in  a  few  moments  is  found  frozen.     As  the  pressure  of  the 
air  is  taken  off,  evaporation  proceeds  with  increased  rapidity,  and 
the  requisite  heat  for  this  change  of  condition  can  be  taken  only 
from  the  dish  of  water.     But  the  atmosphere  of  vapor  retards  the 
process  by  its  pressure ;  hence  the  sulphuric  acid  is  placed  in  the 
receiver,  so  as  to  seize  upon  the  vapor  as  fast  as  formed,  and  thus 
render  the  vacuum  more  complete.     The  water  is  frozen  by  giving 
up  its  heat  to  become  latent  in  the  vapor,  so  rapidly  formed ;  but 
when  this  vapor  becomes  liquid  again  in  combining  with  the  acid, 
the  same  heat  reappears  in  raising  the  temperature  of  the  acid. 

Thin  cakes  of  ice  may  sometimes  be  procured,  even  in  the  hot- 
test climates,  by  the  evaporation  of  water  in  broad  shallow  pans 
under  the  open  sky,  where  radiation  by  night  aids  in  reducing 
the  temperature.  The  pans  should  be  so  situated  as  to  receive  the 
least  possible  heat  by  conduction. 

544.  Spheroidal  Condition. — When  a  little  water  is  placed 
in  a  red-hot  metallic  cup,  instead  of  boiling  violently,  and  disap- 
pearing in  a  moment,  as  might  be  expected,  it  rolls  about  quietly 


STEAM. 


325 


in  the  shape  of  an  oblate  spheroid,  and  wastes  very  slowly.  So 
drops  of  water,  falling  on  the  horizontal  surface  of  a  very  hot  stove, 
are  not  thrown  off  in  steam  and  spray  with  a  loud  hissing  sound, 
as  they  are  when  the  stove  is  only  moderately  heated,  but  roll  over 
the  surface  in  balls,  slowly  diminishing  in  size  till  they  disappear. 
In  such  cases,  the  water  is  said  to  be  in  the  spheroidal  state. 
Not  being  in  contact  with  the  metal,  it  assiwaes  the  shape  of  an 
oblate  spheroid,  in  obedience  to  its  own  molecular  attractions  and 
the  force  of  gravity,  as  small  masses  of  mercury  do  on  a  table. 
The  reason  why  the  water  does  not  touch  the  hot  metal  is,  that 
the  heat  causes  a  coat  of  vapor  to  be  instantly  formed  about  the 
drop,  on  which  it  rests  as  on  an  elastic  cushion ;  and  as  the  vapor 
is  a  poor  conductor  of  heat,  further  evaporation  procee4s  very 
slowly.  It  is  easily  seen  that  the  spheroid  does  not  touch  the 
metal,  by  so  arranging  the  experiment  that  a  beam  of  light  may 
shine  horizontally  upon  the  drop,  and  cast  its  shadow  completely 
separated  from  that  of  the  hot  plate  below  it,  as  in  Fig.  293. 

FIG.  293. 


If  the  heated  surface  is  cooling,  the  temperature  may  become 
so  low  that  the  drop  at  length  touches  it,  when  in  an  instant  vio- 
lent ebullition  takes  place,  and  the  water  quickly  disappears  in 
vapor. 


CHAPTER    IY. 

STEAM.— THE  STEAM-ENGINE.— MECHANICAL  EQUIVALENT 
OF  HEAT. 

545.  Thermal  Force  in  Steam.— It  has  been  already  noticed 
that  while  water  is  heated,  and  especially  while  it  is  converted  into 
steam  by  boiling,  the  heat  apparently  lost  is  so  much  force  treas- 
ured up  ready  for  use,  as  truly  as  when  strength  is  expended  in 
lifting  great  weights,  which  by  their  descent  can  do  the  work  de- 


326 


HEAT. 


sired.  In  modern  engineering,  the  force  of  steam  is  employed 
more  extensively,  and  for  more  varied  purposes,  than  any  other. 
Every  steam-engine  is  a  machine  for  transforming  the 'internal 
motion  of  heated  steam  into  some  of  the  visible  forms  of  motion. 

546.  Tension  cf  Steam. — When  steam  is  formed  by  boiling 
water  in  the  open  air,  its  tension  is  equal  to  that  of  the  air,  and 
therefore  ordinarily  about  fifteen  pounds  to  the  square  inch.  But 
when  it  is  formed  in  a  tight  vessel,  so  that  it  cannot  expand,  as 
the  temperature  of  the  water  is  raised  the  tension  is  increased  in 
a  much  greater  ratio;  because  the  same  steam  has  greater  tension 
at  a  higher  temperature,  and  besides  this,  new  steam  is  continually 
added.  The  following  table  gives  the  temperature  for  successive 
atmospheres  of  tension: 


Atmospheres. 
I  .      . 

Degrees  of 
Temperature. 

.    212 

Atmospheres. 

Degrees 
Temperat 

.    367 

2  .... 

.    2CI 

12        .       .       . 

.    ^7/t 

T.   . 

.    27C 

I  "? 

^81 

I  A. 

-287 

C   . 

.    3O7 

1C 

6  .    .    .    . 

.    ^2O 

16     .     .     . 

7  . 

.    3^2 

17 

8  .     .     .     . 

18     .     .     . 

.    AOQ 

0   . 

.    2CI 

IQ 

•    4-14. 

.   .^Q 

.    4.18 

It  is  seen  by  the  above  table  that  thirty-nine  degrees  of  heat 
are  needed  to  add  the  second  atmosphere  of  tension,  and  that  the 
number  diminishes  constantly,  so  that  only  four  degrees  are  re- 
quired to  add  the  twentieth  atmosphere. 

In  the  formation  of  ordinary  steam  at  212°,  one  cubic  inch  of 
water  is  expanded  to  about  1,700  cubic  inches  of  steam,  or  nearly 
a  cubic  foot.  At  higher  temperatures,  the  volume  diminishes 
nearly  as  fast  as  the  temperature  increases. 

547.  The  Steam-engines  of  Savery  and  Newcomen.— 

The  only  steam-engines  that  were  at  all  successful  before  the  great 
improvements  made  by  Watt,  were  the  engine  of  Savery  and  that 
of  Newcomen.  No  other  purpose  was  proposed  by  either  than  that 
of  removing  water  from  mines. 

In  the  engine  of  Savery,  steam  was  made  to  raise  water  by  act- 
ing on  it  directly,  and  not  through  the  intervention  of  machinery. 
First,  the  steam  in  a  vessel  was  condensed  by  cold  water  flowing 
over  the  outside,  and  the  atmosphere  raised  water  into  the  ex- 
hausted vessel  by  its  pressure.  In  the  next  place,  steam  was  let 


THE    STEAM-ENGINE.  307 

into  the  vessel,  and  by  its  tension  forced  the  water  out,  and  raised 
it  still  higher.  The  water  raised  by  each  part  of  this  operation 
was  prevented  from  returning  by  a  valve,  as  in  a  forcing  pump. 

Newcomen  employed  steam  in  a  very  different  way,  namely,  as 
a  power  to  work  a  common  pump.  The  pump  rod  was  attached 
to  one  end  of  a  working  beam,  and  to  the  other  end  of  the  same 
was  attached  the  rod  of  the  steam  piston,  which  moved  steam-tight 
in  a  cylinder.  The  end  of  the  beam  next  to  the  pump  was  made 
heavy  enough  to  keep  the  steam  piston  at  the  top  of  the  cylinder, 
when  no  force  was  applied.  The  space  beneath  the  piston  being 
filled  with  steam,  a  little  cold  water  was  injected,  the  steam  con- 
densed, and  the  piston  forced  down  by  the  weight  of  the  air  on 
the  top  of  it.  Then,  as  soon  as  steam  was  admitted  again  below, 
though  having  no  greater  tension  than  the  atmosphere,  the  piston 
was  drawn  up  by  the  weight  of  the  opposite  end  of  the  beam.  Since 
the  water  was  raised  directly  by  the  weight  of  the  atmosphere, 
after  the  steam  had  given  it  opportunity  to  act,  this  invention  of 
Newcomen  was  called  the  atmospheric  engine. 

But  in  neither  of  these  methods  was  steam  used  economically 
as  a  power.  The  movements  in  both  cases  were  sluggish,  and  a 
large  part  of  the  force  was  wasted,  because  the  steam  was  com- 
pelled to  act  upon  a  cold  surface,  which  condensed  it  before  its 
work  was  done. 

548.  The  Steam-engine  of  Watt— Steam  did  not  give 
promise  of  being  essentially  useful  as  a  power  till  Watt,  in  the 
year  1760,  made  a  change  in  the  atmospheric  engine,  which  pre- 
vented the  great  waste  of  force.  Newcomen  introduced  the  cold 
water  which  was  to  condense  the  steam  into  the  steam  cylinder 
itself;  and  the  cylinder  must  be  cooled  to  a  temperature  below 
100°,  else  there  would  be  steam  of  low  tension  to  retard  the  de- 
scent of  the  piston.  But  when  the  piston  was  to  be  raised,  the 
cylinder  must  be  heated  again  to  212°,  in  order  that  the  admitted 
steam  might  balance  the  pressure  of  the  air. 

In  the  engine  of  Watt,  the  steam  is  condensed  in  a  separate 
vessel  called  the  condenser.  The  steam  cylinder  is  thus  kept  at 
the  uniform  temperature  of  the  steam.  In  the  first  form  which 
he  gave  to  his  engine,  he  so  far  copied  the  atmospheric  engine  as 
to  allow  the  piston,  after  being  pressed  down  by  steam,  to  be  raised 
again  by  the  load  on  the  opposite  end  of  the  great  beam,  while  the 
steam  circulates  freely  below  and  above  the  piston.  This  was 
called  the  single-acting  engine,  and  might  be  successfully  used  for 
the  only  use  to  which  any  steam-engine  was  as  yet  applied, 
namely,  pumping  water  from  mines.  But  he  almost  immediately 
introduced  the  change  by  which  the  whole  force  of  the  steam  was 


323 


HEAT. 


FIG.  294. 


brought  to  act  on  the  upper  and  the  under  side  of  the  piston.  It 
thus  became  double-acting,  and  the  steam  force  was  no  longer 
intermittent. 

549.  The  Double-acting  Engine.— Let  S  (Fig.  294)  be  the 
steam  cylinder,  P  the  piston,  A  the  piston  rod,  passing  with 
steam-tight  joint  through  the  top 
of  the  cylinder,  C  the  condenser, 
kept  cold  by  the  water  of  the  cis- 
tern 6r,  B  the  steam  pipe  from 
the  boiler,  K  the  eduction  pipe, 
which  opens  into  the  valve  chest 
at  0,  D  D  the  D- valve,  E  F  the 
openings  from  the  valve-chest  into 
the  cylinder.  As  the  D-valve  is 
situated  in  the  figure,  the  steam 
can  pass  through  B  and  JS'into 
the  cylinder  below  the  piston, 
while  the  steam  above  the  piston 
can  escape  by  F  through  0  and  K 
to  the  condenser,  where  it  is  con- 
densed as  fast  as  it  enters ;  so  that 
in  an  instant  the  space  above  the 
piston  is  a  vacuum,  while  the 
whole 
erted 


is  a  vacuum, 
force  of  the  steam  is  ex- 
the   under  side.     The 


on 


piston  is  therefore  driven  upward  without  any  force  to  oppose  it. 
But  before  it  reaches  the  top,  the  D-valve,  moved  by  the  machin- 
ery, begins  to  descend,  and  shut  off  the  steam  from  E  and  admit  it 
to  F,  and,  on  the  other  hand,  to  shut  F  from  the  eduction  pipe  0, 
and  open  E  to  the  same.  The  steam  will  then  press  on  the  top 
of  the  piston,  and  there  will  be  a  vacuum  below  it,  so  that  the  pis- 
ton descends  with  the  whole  force  of  the  steam,  and  without 
resistance.  To  render  the  condensation  more  sudden,  a  little  cold 
water  is  thrown  into  the  condenser  at  each  stroke  through  the 
pipe  H. 

550.  The  Low-pressure  Engine. — The  principle  of  the  low- 
pressure  engine  is  illustrated  by  the  figure  and  description  of  the 
preceding  article.  But  the  condensing  apparatus  of  this  kind  of 
engine  requires  many  other  parts,  most  of  which  are  presented  in 
Fig.  295.  C  is  the  steam  cylinder ;  R  the  rod  connecting  its  pis- 
ton with  the  end  of  the  working  beam,  not  represented ;  A  the 
steam  pipe  and  throttle  valve ;  B  B  the  D-valve ;  D  D  the  educ- 
tion pipe,  leading  from  the  valve  chest  to  the  condenser  E\  G  G 
the  cold  water  surrounding  the  condenser;  J^the  air-pump,  which 


THE    LOW-PRESSURE    ENGINE. 


329 


keeps  the  condenser  clear  of  air,  steam,  and  water  of  condensation ; 
/  the  hot  well,  in  which  the  water  of  condensation  is  deposited  bv 
the  air-pump;  K  the  hot-water  pump,  which  forces  the  water 


FIG.  295. 


in  the  hot  well  through  L  to  the  boiler ;  H  the  cold-water  pump,  by 
which  water  is  brought  to  the  cistern  G  G',  the  rods  of  all  the 
pumps,  F,  K,  and  H,  are  moved  by  the  working  beam;  P  the 
fly-wheel ;  M  the  crank  of  the  same,  JV  the  connecting-rod,  by 
which  the  working  beam  conveys  motion  to  the  fly-wheel ;  Q  the 
excentric  rod,  by  which  the  D-valve  is  moved ;  0  0  the  governor, 
which  regulates  the  throttle  valve  in  the  steam  pipe  A. 

551.  Steam- Valves. — What  are  commonly  called  valves  in 
steam  machinery  are  not  strictly  such,  since  they  are  not  opened 
by  the  pressure  of  a  fluid  in  one  direction,  and  closed  by  a  pres- 
sure in  the  opposite  direction.  On  the  contrary,  they  are  opened 
and  shut  by  the  action  of  an  eccentric  cam  on  the  principal  axis. 
The  puppet  valve  is  the  frustum  of  a  cone,  fitting  into  a  conical 
socket,  and  opens  the  pipe  by  being  raised.  The  sliding  valve  does 
not  rise,  but  slides  over  the  aperture.  The  rotary  valve,  like  the 
common  stop-cock,  is  a  cylinder  lying^  across  the  pipe,  and  having 
an  aperture  through  it,  so  that  by  a  quarter  revolution  it  opens  or 
shuts  the  pipe.  The  throttle  valve,  like  the  damper  of  a  stove-pipe, 
is  a  partition  in  the  pipe,  turning  on  an  axis,  so  as  to  lie  crosswise 
or  lengthwise.  The  D-valve,  so  called  from  its  form,  is  a  sliding 
valve  which  takes  the  place  of  four  valves  in  the  earlier  engines, 


330  HEAT. 

connecting  both  the  top  and  the  bottom  of  the  steam  cylinder 
with  the  boiler  and  with  the  condenser. 

There  is  much  economy  of  fuel  and  saving  of  wear  in  the 
machinery,  arising  from  the  proper  adjustment  of  the  valves.  If 
the  steam  enters  the  cylinder  during  the  whole  length  of  a  stroke 
of  the  piston,  its  motion  is  accelerated ;  and  is  therefore  swiftest 
at  the  instant  before  being  stopped ;  thus  the  machinery  receives 
a  violent  shock.  If  the  valve  is  adjusted  to  cut  off  the  steam  when 
the  piston  has  made  one-third  or  one-half  of  its  stroke,  the  dimin- 
ishing tension  may  exert  about  force  enough,  during  the  remain- 
ing part,  to  keep  up  a  uniform  motion.  The  cut-off,  however, 
should  be  regulated  in  each  engine,  according  to  friction  and 
other  obstructions. 

552.  High-Pressure  Engine. — The  engine  of  "Watt,  already 
described,  is  properly  called  a  low-pressure  engine,  because  the 
steam,  having  a  vacuum  on  the  opposite  side  of  the  piston,  works 
at  the  least  possible  tension.    For  many  purposes,  especially  those 
of  locomotion,  it  is  advantageous  to  dispense  with  the  large  weight 
and  bulk  of  machinery  necessary  for  condensation,  and  do  the 
work  with  steam  of  a  higher  tension.     In  Fig.  295,  if  the  con- 
denser, cistern,  and  all  the  pumps  are  removed,  then  the  steam  is 
discharged  from  E  and  F  at  each  stroke  into  the  air.     Therefore 
the  steam  in  that  part  of  the  cylinder  which  is  open  to  the  air, 
will  have  a  tension  of  15  Ibs.  per  inch ;  and,  consequently,  the 
steam  on  the  opposite  side  of  the  piston  must  have  a  tension  15 
Ibs.  per  inch  greater  than  before,  in  order  to  do  the  same  work. 

553.  Applications  of  Steam  Power. — For  more  than  half 
a  century,  the  only  use  of  the  steam-engine  was  to  work  the  water 
pumps  of  the  English  mines.     But  the  genius  of  Watt  has  ren- 
dered it  available  for  nearly  every  purpose  which  requires  the  use 
of  machinery.     Every  description  of  machine,  for  the  heaviest  and 
the  lightest  operation,  may  have  a  steam-engine  for  its  prime 
mover.    Near  the  beginning  of  the  present  century,  it  began  to  be 
used  for  locomotion  on  land  and  water ;  and  at  the  present  day, 
both  traveling  and  the  transportation  of  merchandise  are  princi- 
pally accomplished  by  means  of  steam. 

554.  Estimation  of  Steam  Power.— It  is  customary  to  ex- 
press the  power  of  a  steam-engine  by  comparing  it  with  the  num- 
ber of  horses  whose  strength  it  equals.    In  making  this  compari- 
son, Watt  took  as  a  measure  of  one  horse-power,  the  ability  to  raise 
2,000,000  Ibs.  through  the  height  of  one  foot  in  an  hour ;  or  2,000,- 
000  foot-pounds  per  hour.    It  is  obviously  immaterial  what  the 
respective  factors  for  feet,  and  for  pounds,  are,  if  the  product  only 


JOULE'S    EQUIVALENT.  331 

equals  2,000,000.  For  example,  2,000  Ibs.  through  1,000  feet, 
or  5,000  Ibs.  400  feet  per  hour,  &c.,  is  equal  to  one  horse-power. 
It  is  found  that  the  available  force  of  one  cubic  foot  of  water,  when 
changed  to  steam,  is  about  equal  to  2,000,000  foot-pounds ;  that 
is,  to  one  horse-power.  Hence,  an  engine  of  fifty  horse-power  is 
one  which  can  change  fifty  cubic  feet  of  water  into  steam  in  one 
hour. 

555.  Mechanical  Equivalent  of  Heat. — In  all  cases  in 
which  mechanical  force  produces  heat,  and  again  in  all  those  in 
which  heat  produces  visible  motion,  careful  experiment  proves 
that  heat  and  mechanical  force  may  each  be  made  a  measure  of 
the  other.  Forces  of  any  kind  may  be  compared,  by  observing 
the  weights  which  they  will  lift  through  a  given  distance.  The 
mechanical  equivalent  of  heat  (commonly  called,  from  the  name  of 
an  English  experimenter,  Joule's  equivalent)  is  given  in  the  fol- 
lowing statement : 

The  force  required  to  heat  one  pound  of  water  one  degree  F.,  is 
equal  to  that  which  would  lift  772  pounds  the  distance  of  one  foot, 
or  is  equal  to  772  foot-pounds.  . 

The  force  requisite  to  raise  one  pound  of  water  1°  F.,  is  some- 
times called  the  thermal  unit  (Art.  535),  and  all  forces  may  be 
brought  to  this  as  a  standard  of  comparison.  Thus,  one  horse- 
power (2,000,000  foot-pounds  per  hour)  is  2,590  thermal  units  per 
hour,  or  about  43  per  minute. 

Since  a  force  of  772  foot-pounds  is  expended  in  heating  a 
pound  of  water  1°  F.,  therefore  to  heat  the  same  from  32°  to  212° 
requires  a  force  of  138,960  foot-pounds ;  and  to  change  the  same 
pound  of  water  into  steam  of  atmospheric  tension  requires  an  ad- 
ditional force  of  746,900  foot-pounds  (Art.  539). 


CHAPTER    Y. 

TEMPERATURE  OF  THE  ATMOSPHERE.— MOISTURE  OF  THE  AT- 
MOSPHERE.—DRAFT  AND  VENTILATION. 

556.  Manner  in  which  the  Air  is  Warmed. — The  space 
through  which  the  earth  moves  around  the  sun  is  intensely  cold, 
probably  75°  below  zero.  And  the  one  or  two  hundred  miles  of 
height  occupied  by  the  atmosphere  is  too  cold  for  animal  or  vege- 
table life,  except  the  lowest  stratum,  three  or  four  miles  in  thick- 
ness. This  portion  receives  its  heat  mainly  by  convection.  The 


332  HEAT. 

radiated  heat  of  the  sun  passes  through  the  air,  warming  it  but 
little,  and  on  reaching  the  earth  is  partly  absorbed  by  it.  The  air 
lying  in  contact  with  the  earth,  and  thus  becoming  warmed,  grows 
lighter  and  rises,  while  colder  portions  descend  and  are  warmed  in 
their  turn.  So  long  as  the  sun  is  shining  on  a  given  region  of  the 
earth,  this  circulation  is  going  on  continually.  But  the  heated 
air  which  rises  is  expanded  by  diminished  pressure,  and  thus 
cooled.  Hence  the  circulation  is  limited  to  a  very  few  miles  next 
to  the  earth. 

557.  Limit  of  Perpetual  Frost. — At  a  moderate  elevation, 
even  in  the  hottest  climate,  the  temperature  of  the  air  is  always 
as  low  as  the  freezing  point.     Hence  the  permanent  snow  on  the 
higher  mountains  in  all  climates.    The  limit  at  the  equator  is 
about  three  miles  high,  and  with  many  local  exceptions  it  de- 
scends each  way  to  the  polar  regions,  where  it  is  very  near  the 
earth.     The  descent  is  more  rapid  in  the  temperate  than  in  the 
torrid  or  frigid  zones. 

558.  Isothermal  Lines. — These  are  imaginary  lines  on  each 
hemisphere,  through  all  those  points  whose  mean  annual  tempera- 
ture is  the  same.     At  the  equator,  the  mean  temperature  is  about 
82°,  and  it  decreases  each  way  toward  the  poles,  but  not  equally 
on  all  meridians.     Hence  the  isothermal  lines  deviate  widely  from 
parallels  of  latitude.     Their  irregularities  are  due  to  the  difference 
between  land  and  water,  in  absorbing  and  communicating  heat, 
to  the  various  elevations  of  land,  especially  ranges  of  mountains, 
to  ocean  currents,  &c.     In  the  northern  hemisphere,  the  isother- 
mal lines,  in  passing  westward  round  the  earth,  generally  descend 
toward  the  equator  in  crossing  the  oceans,  and  ascend  again  in 
crossing  the  continents.      For  example,  the  isothermal  of  50°, 
which  passes  through  China  on  the  parallel  of  44°,  ascends  in 
crossing  the  eastern  continent,  and  strikes  Brussels,  lat.  51°  ;  and 
then  on  the  Atlantic,  descends  to  Boston,  lat.  42°,  whence  it  once 
more  ascends  to  the  N.  W.  coast  of  America.    The  lowest  mean 
temperature  in  the  northern  hemisphere  is  not  far  from  zero,  but 
it  is  not  situated  at  the  north  pole.     Instead  of  this,  there  are  two 
poles  of  greatest  cold,  one  on  the  eastern  continent,  the  other  on 
the  western,  near  20°  from  the  geographical  pole.     There  are  indi- 
cations, also,  of  two  south  poles  of  maximum  cold. 

559.  Moisture  of  the  Atmosphere.— By  the  heat  of  the 
sun  all  the  waters  of  the  earth  form  above  them  an  atmosphere  of 
vapor,  or  invisible  moisture,  having  more  or  less  extent  and  ten- 
sion, according  to  several  circumstances.    Even  ice  and  snow,  at 
the  lowest  temperatures,  throw  off  some  vapor.    A  gaseous  body 


MEASURE     OF    VAPOR.  333 

diffuses  itself  by  its  force  of  tension,  whether  another  gas  occupies 
the  same  space  or  not ;  that  is,  the  particles  of  one  do  not  exert 
a  perceptible  attraction  or  repulsion  on  those  of  the  other,  but 
each  is  a  vacuum  to  the  other,  except  so  far  as  it  obstructs  its 
movements.  Therefore,  at  a  given  temperature,  there  can  exist  an 
atmosphere  of  vapor  of  the  same  height  and  tension,  whether 
there  is  an  atmosphere  of  oxygen  and  nitrogen  or  not.  Vapor, 
then,  is  not  strictly  suspended  in  the  air,  or  dissolved  by  it,  but 
exists  independently.  And  yet  it  is  by  no  means  always  true  that 
there  is  actually  the  same  tension  of  yapor  as  there  would  be  if  it 
existed  alone,  because  of  the  time  required  for  the  formation  of 
vapor,  on  account  of  mechanical  obstruction  presented  by  the  air; 
whereas,  if  no  air  existed,  the  vapor  would  form  almost  instantly. 
It  is  on  the  same  account  that  water  will  boil  in  a  vacuum  at  72°, 
but  under  the  pressure  of  the  air  must  be  heated  to  212°. 

560.  Temperature  and  Tension  of  Vapor. — The  degree 
of  tension  of  vapor  forming  without  obstruction,  depends  on  its 
temperature,  but  varies  far  more  rapidly ;  increasing  pretty  nearly 
in  a  geometrical  ratio,  while  the  heat  increases  arithmetically. 
Hence,  if  vapor  should  receive  its  full  increment  of  tension,  while 
the  thermometer  rises  10  degrees  from  80°  to  90°,  a  vastly  greater 
quantity  would  be  added  than  when  it  rises  10  degrees  from  40° 
to  50°.     On  the  contrary,  if  vapor  is  at  its  full  tension  in  each 
case,  much  more  water  will  be  precipitated  in  cooling  from  90°  to 
80°  than  from  50°  to  40°. 

561.  Dew-point. — This  is  the  temperature  at  which  vapor, 
in  a  given  case,  is  precipitated  into  water  in  some  of  its  forms. 
If  there  was  no  air,  the  dew-point  would  always  be  the  same  as 
the  existing  temperature ;   since  lowering  the  temperature  in  the 
least  degree  would  require  a  diminished  tension  or  quantity  of 
vapor,  some  must  therefore  be  condensed  into  water.     But  in  the 
air  the  tension  may  not  be  at  its  full  height,  and  therefore  the 
temperature  may  need  to  be  reduced  several  degrees  before  precip- 
itation will  take  place.    A  comparison  of  the  temperature  with 
the  dew-point  is  one  of  the  methods  employed  for  measuring  the 
humidity  of  the  air. 

562.  Measure  of  Vapor. — The  measure  of  the  vapor  exist- 
ing at  a  given  time,  is  expressed  by  two  numbers,  one  indicating 
its  tension, — i.  e.,  the  height  of  the  column  of  mercury  which  it 
will  sustain ;  the  other,  humidity, — i.  e.,  its  quantity  per  cent.,  as 
compared  with  the  greatest  possible  amount  at  that  temperature. 
Thus,  tension  =  0.6,  humidity  —  83,  signifies  that  the  quantity  of 
vapor  is  sufficient  to  support  six-tenths  of  an  inch  of  mercury,  and 


334  HEAT. 

is  83  hundredths  of  the  quantity  which  could  exist  at  that  tem- 
perature. The  greatest  tension  possible  at  zero,  is  0.04 ;  at  the 
freezing  point,  0.18  ;  at  80°,  1.0.  At  the  lowest  natural  tempera- 
tures, the  maximum  tension  is  doubled  every  12°  or  14° ;  at  the 
highest,  every  21°  or  22°. 

583.  Hygrometers. — This  is  the  name  usually  given  to  in- 
struments intended  for  measuring  the  moisture  of  the  air.     But 
the  one  most  used  of  late  years  is  called  the  psychr ometer,  which 
gives  indication  of  the  amount  of  moisture  by  the  degree  of  cold 
produced  in  evaporation ;  for  evaporation  is  more  rapid,  and  there- 
fore the  cold  occasioned  by  it  the  greater,  according  as  the  air  is 
drier.    The  psychrometer  consists  of  two  thermometers,  one  hav- 
ing its  bulb  covered  with  muslin,  and  moistened  before  the  ob- 
servation.    The  wet-bulb  thermometer  will  ordinarily  indicate  a 
lower  temperature  than  the  dry-bulb  ;  if,  in  a  given  case,  they  read 
alike,  the  humidity  is  100.    The  instrument  is  accompanied  by 
tables,  giving  tension  and  humidity  for  any  observation. 

584.  Precipitations   of  Moisture. — Whenever  the  air  is 
cooled  below  the  dew-point,  a  part  of  the  vapor  is  deposited  in  the 
liquid  or  solid  form.     The  precipitations  occur  under  various  con- 
ditions, and  receive  the  following  names :  dew,  frost,  fog,  cloud, 
rain,  mist,  hail,  sleet,  and  snow. 

565.  Dew. — Frost. — The  deposition  called  dew  takes  place  on 
the  surface  of  bodies,  by  which  the  air  is  cooled  below  its  dew- 
point.  It  is  at  first  in  the  form  of  very  small  drops,  which  unite 
and  enlarge  as  the  process  goes  on.  Dew  is  formed  in  the  even- 
ing or  night,  when  the  surfaces  of  bodies  exposed  to  the  sky 
become  cold  by  radiation.  As  soon  as  their  temperature  has 
descended  to  the  dew-point,  the  stratum  of  air  contiguous  to  them 
deposits  moisture,  and  continues  to  do  so  more  and  more  as  the 
cold  increases. 

Of  two  bodies  in  the  same  situation,  that  will  receive  most 
dew  which  radiates  most  rapidly.  Many  vegetable  leaves  are 
good  radiators,  and  receive  much  dew.  Polished  metal  is  a  poor 
radiator,  and  ordinarily  has  no  dew  deposited  on  it. 

Sometimes,  however,  good  radiators  have  little  dew,  because 
they  are  so  situated  as  to  obtain  heat  nearly  as  fast  as  they  radiate 
it.  Dew  is  rarely  formed  on  a  bed  of  sand,  though  it  is  a  good 
radiator,  because  the  upper  surface  gets  heat  by  conduction  from 
the  mass  below.  Dew  is  not  formed  on  water,  because  the  upper 
stratum  sinks  and  gives  place  to  warmer  ones. 

Bodies  most  exposed  to  the  open  sky,  other  things  being  equal, 
have  most  dew  precipitated  on  them.  This  is  owing  to  the  fact, 


CLOUD,  335 

that  in  such  circumstances,  they  have  no  return  of  heat  either  by 
reflection  or  radiation.  If  a  body  radiates  its  heat  to  a  building, 
a  tree,  or  a  cloud,  it  also  gets  some  in  return,  both  reflected  and 
radiated.  Hence,  little  dew  is  to  be  expected  in  a  cloudy  night, 
or  on  objects  surrounded  by  high  trees  and  buildings. 

Wind  is  unfavorable  to  the  formation  of  dew,  because  it  min- 
gles the  strata,  and  prevents  the  same  mass  from  resting  long 
enough  on  the  cold  body  to  be  cooled  down  to  the  dew-point. 

When  the  radiating  body  is  cooled  below  the  freezing  point, 
the  water  deposited  takes  the  solid  form  in  fine  crystals,  and  is 
called  frost.  Frost  will  often  be  found  on  the  best  radiators,  or 
those  exposed  to  the  open  sky,  when  only  dew  is  found  elsewhere. 

566.  Fog, — This  form  of  precipitation  consists  of  very  small 
globules  of  water  sustained  in  the  lower  strata  of  the  air.    Fog 
occurs  most  frequently  over  low  grounds  and  bodies  of  water, 
where  the  humidity  is  likely  to  be  great.    If  air  thus  humid  mixes 
with  air  cooled  by  neighboring  land,  even  of  less  humidity,  there 
will  probably  be  more  vapor  than  can  exist  at  the  intermediate 
temperature,  for  the  reason  mentioned  in  Art.  560.     The  case  may 
be  illustrated  thus.    Let  two  masses  of  air  of  equal  volumes  be 
mixed,  the  temperature  of  one  being  40°,  the  other  60°  ;  and  each 
containing  vapor  at  the  highest  tension.    Then  the  mixture  will 
have  the  mean  temperature  of  50°,  and  the  vapor  of  the  mixture 
will  also  be  the  arithmetical  mean  between  that  of  the  two  masses. 
But,  according  to  the  law  (Art.  560),  the  vapor  can  only  have  a 
tension  which  is  nearly  a  geometrical  mean  between  the  two,  and 
that  is  necessarily  lower  than  the  arithmetical  mean ;  hence  the 
excess  must  be  precipitated.    If  8  Ibs.  of  vapor  were  in  one  volume 
and  18  Ibs.  in  the  other,  an  equal  volume  of  the  mixture  would 
have  -|-  (8  +  18)  =  13  Ibs.  of  moisture ;  but  at  the  mean  tempera, 
ture  of  50°,  only  Vs  x  18  =  12  Ibs.  could  exist  as  vapor ;   there- 
fore one  pound  must  be  precipitated.    And  even  if  one  of  the 
masses  had  a  humidity  somewhat  below  100,  still  some  precipita- 
tion is  likely  to  take  place. 

567.  Cloud. — The  same  as  fog,  except  at  a  greater  elevation. 
Air  rising  from  heated  places  on  the  earth,  and  carrying  vapor 
with  it,  is  likely  to  meet  with  masses  much  colder  than  itself,  and 
depositions  of  moisture  are  therefore  likely  to  take  place.     Mount- 
ain-tops are  often  capped  with  clouds,  when  all  around  is  clear. 
This  happens  when  lower  and  warmer  strata  are  driven  over  them, 
and  thus  cooled  below  the  dew-point.    The  same  air,  as  it  con- 
tinues down  the  other  side,  takes  up  its  vapor  again,  and  is  as 
transparent  as  it  was  before  ascending.    A  person  on  the  summit 


336  HEAT. 

perceives  a  chilly  fog  driving  by  him,  but  the  fog  was  an  invisible 
vapor  a  few  minutes  before  reaching  him,  and  returns  to  the  same 
condition  soon  after  leaving  him.  The  cloud  rests  on  the  mount- 
ain ;  but  all  the  particles  which  compose  it  are  swiftly  crossing 
over.  Clouds  are  often  above  the  limit  of  perpetual  frost ;  they 
then  consist  of  crystals  of  ice. 

568.  Classification  of  Clouds. — The  aspects  of  clouds  are 
various,  and  depend  in  some  measure  at  least  on  the  circumstances 
of  their  formation.     The  usual  classification  is  the  following  : 

1.  Cirrus. — This  cloud  is  fibrous  in  its  appearance,  like  hair  or 
flax,  sometimes  straight,  sometimes  bent,  and  frequently  at  one 
end  is  gathered  into  a  confused  heap  of  fibers.    The  cirrus  is  high, 
and  often  consists  of  frozen  particles,  even  in  summer. 

2.  Ctimulus.—This  consists  of  compact  rounded  heaps,  which 
often  resemble  mountain-tops  covered  with  snow.    This  form  of 
cloud  is  confined  mostly  to  the  summer  season  ;  it  usually  begins 
to  form  after  the  sun  rises,  and  to  disappear  before  it  sets,  and  is 
rarely  seen  far  from  land.    The  cumulus  is  generally  not  so  high 
as  the  cirrus. 

3.  Strains. — Sheets  or  stripes  of  cloud,  sometimes  overspread- 
ing the  whole  sky,  or  as  a  fog  covering  the  surface  of  the  earth  or 
water.    The  stratus  is  the  most  common,  and  usually  lies  lowest 
in  the  air. 

4.  5,  6.   Cirro-cumulus,  cirro-stratus,  cumulo-straius. — Inter- 
mediate or  combined  forms. 

7.  Nimbus. — A  cloud,  which  forms  so  fast  as  to  fall  in  rain  or 
snow,  is  called  by  this  name. 

569.  Rain,  Mist. — "Whether  the  precipitated  moisture  has  the 
form  of  cloud  or  rain,  depends  on  the  rapidity  with  which  precip- 
itation takes  place.    If  currents  of  air  are  in  rapid  motion,  if  the 
temperature  of  masses,  brought  into  contact  by  this  motion,  are 
widely  different,  and  if  their  humidity  is  at  a  high  point,  the  vapor 
will  be  precipitated  so  rapidly,  that  the  globules  will  touch  each 
other,  and  unite  into  larger  drops,  which  cannot  be  sustained. 
Globules  of  fog  and  cloud,  however,  are  specifically  as  heavy  as 
drops  of  rain ;  but  they  are  sustained  by  the  slightest  upward 
movements  of  the  air,  because  they  have  a  great  surface  compared 
with  their  weight.     A  globule  whose  diameter  is  100  times  less 
than  that  of  a  drop  of  rain,  meets  with  100  times  more  obstruction 
in  descending,  since  the  weight  is  diminished  a  million  times 
(TJ0)8,  and  the  surface  only  ten  thousand  times  (T  Jo)2.    So  the 
dust  of  even  heavy  minerals  is  sustained  in  the  air  for  some  time, 
when  the  same  substances,  in  the  form  of  sand,  or  coarse  gravel, 
fall  instantly. 


THEORIES    OF   PRECIPITATION.  337 

Mist  is  fine  rain ;  the  drops  are  barely  large  enough  to  make 
their  way  slowly  to  the  earth. 

570.  Hail,  Sleet,  Snow. — "When  the  air  in  which  rapid  pre- 
cipitation occurs,  is  so  cold  as  to  freeze  the  drops,  hail  is  produced. 
As  hailstones  are  not  usually  in  the  spherical  form  when  they 
reach  the  earth,  it  is  supposed  that  they  are  continually  receiving 
irregular  accretions  in  their  descent  through  the  vapor  of  the  air. 
Hail-storms  are  most  frequent  and  violent  in  those  regions  where 
hot  and  cold  bodies  of  air  are  most  easily  mixed.     Such  mixtures 
are  rarely  formed  in  the  torrid  zone,  since  there  the  cold  air  is  at  a 
great  elevation  ;  in  the  frigid  zone,  no  hot  air  exists  at  any  height; 
but  in  the  temperate  climates,  the  heated  air  of  the  torrid,  and  the 
intensely  cold  winds  of  the  frigid  zone,  may  be  much  more  easily 
brought  together;   and  accordingly,  in  the  temperate  zones  it  is 
that  hail-storms  chiefly  occur.    Even  in  these  climates,  they  are 
not  frequent  except  on  plains  and  in  valleys  contiguous  to  mount- 
ains which  are  covered  with  snow  during  the  summer.       The 
slopes  of  the  mountain  sides  give  direction  to  currents  of  air,  so 
that  masses  of  different  temperature  are  readily  mingled  together. 

Sleet  is  frozen  mist,  that  is,  it  consists  of  very  small  hailstones. 

Snow  consists  of  the  small  crystals  of  frozen  cloud,  united  in 
flakes.  Like  all  transparent  substances,  when  in  a  pulverized 
state,  it  owes  its  whiteness  to  innumerable  reflecting  surfaces.  A 
cloud,  when  the  sun  shines  upon  it,  is  for  the  same  reason  intensely 
white. 

571.  Theories  of  Precipitation. — It  is  probable  that  clouds 
and  rain  are  caused  not  only  by  the  mixing  of  air  of  different  tem- 
peratures, but  also  by  the  changes  which  take  place  in  the  condi- 
tion of  the  air  as  it  ascends. 

In  the  lower  strata,  the  air  is  about  one  degree  colder  for  every 
300  feet  of  elevation.  If,  therefore,  a  mass  of  air  is  transferred 
from  the  surface  of  the  earth  to  a  height  in  the  atmosphere,  it  will 
be  cooled  to  the  temperature  of  the  stratum  which  it  reaches  ;  not 
principally  by  giving  off  its  heat,  but  by  expanding,  and  thus 
having  its  own  heat  reduced  by  being  diffused  through  a  larger 
space.  ISTow,  if  the  rising  mass  was  saturated  with  moisture,  this 
moisture  would  begin  at  once  to  be  precipitated  by  the  cooling 
which  it  undergoes  in  consequence  of  expansion.  If,  instead  of 
being  saturated,  its  dew-point  is  a  certain  number  of  degrees  below 
its  temperature,  it  must  ascend  far  enough  to  be  cooled  to  the  dew- 
point,  before  precipitation  of  its  moisture  will  take  place.  Sup- 
pose, for  instance,  the  temperature  at  the  earth  is  70°,  and  the  dew- 
point  is  G5°  ;  then  after  the  warm  air  has  risen  1500  feet  (5  x  300 


338  HEAT. 

ft.),  it  will  become  5°  cooler,  and  contain  all  the  moisture  which 
is  possible  at  that  temperature.  At  that  point  precipitation  be- 
gins, and 'forms  the  base  of  a  cloud.  The  clouds,  called  cumulus, 
which  are  seen  forming  during  many  summer  forenoons,  are  the 
precipitations  of  columns  rising  from  warm  spots  of  earth  so  high 
that  they  are  cooled  below  their  dew-point.  But  the  movement 
and  the  precipitation  do  not  stop  here ;  for,  as  moisture  is  precip- 
itated, its  latent  heat  is  given  off  in  large  quantities,  which  ele- 
vates the  temperature  of  the  mass,  and  causes  it  to  rise  still  higher, 
and  precipitate  still  more  of  its  moisture.  As  it  becomes  rarer, 
it  spreads  laterally,  and  causes  the  cumulus  often  to  assume  the 
overhanging  form  which  distinguishes  that  species  of  cloud. 

572.  Cyclones.— The  late  Mr.  Eedfield  investigated  with  great 
success  the  phenomena  of  violent  storms,  especially  of  Atlantic 
hurricanes,  and  showed  that  they  are  generally,  if  not  always, 
great  whirlwinds,  called  cyclones.    They  usually  take  their  rise  in 
the  equatorial  region  eastward  of  the  West  India  Islands ;  they 
rotate  on  a  vertical  axis,  advancing  slowly  to  the  northwest,  until 
they  approach  the  coast  of  the  United  States  near  the  latitude  of 
30°,  and  then  gradually  veer  to  the  northeast,  running  nearly  par- 
allel to  the  American  coast,  and  finally  spend  themselves  in  the 
northern  Atlantic.    Their  rotary  motion  is  always  in  one  direc- 
tion, namely,  from  the  east  through  the  north  to  the  west,  or 
against  the  sun.     This  motion  is  also  far  more  violent,  especially 
in  the  central  parts  of  the  storm,  than  the  progressive  motion. 
The  rotary  motion  may  amount  to  50  or  100  miles  per  hour,  while 
the  forward  motion  of  the  storm  is  not  more  than  15  or  20  miles. 

In  the  southern  hemisphere  also,  cyclones  occur,  having  a  pro- 
gressive and  a  rotary  motion,  both  symmetrical  with  those  of  the 
northern  cyclones.  On  the  axis  they  revolve  with  the  sun,  not 
against  it ;  and  they  first  advance  toward  the  southwest,  and  grad- 
ually veer  toward  the  southeast,  as  they  recede  from  the  equator. 

573.  Draught  of  Flues. — The  effect  of  the  sun's  heat  in 
causing  circulation  of  the  air  has  been  already  considered  (Art. 
289-293).    Similar  movements  on  a  limited  scale  are  produced 
whenever  a  portion  of  the  air  is  heated  by  artificial  means.    Thus, 
the  air  of  a  chimney  is  made  lighter  by  a  fire  beneath  it,  than  a 
column  of  the  outer  air  extending  to  the  same  height.     It  is  there- 
fore pressed  upward  by  the  heavier  external  air,  which  descends 
and  moves  toward  the  place  of  heat.     The  difference  of  weight  in 
the  two  columns  is  greater,  and  therefore  the  draught  stronger,  if 
the  chimney  is  high,  provided  the  supply  of  heat  is  sufficient  to 
maintain  the  requisite  temperature.     Chimneys  are  frequently 


VENTILATION    OF    APARTMENTS.  339 

built  one  or  two  hundred  feet  high  for  the  uses  of  manufactories. 
The  high  fireplaces  and  large  flues  of  former  times  were  unfavora- 
ble for  draught,  both  because  much  cold  air  could  mingle  with 
that  which  was  heated,  and  because  there  was  room  for  external 
air  to  descend  by  the  side  of  the  ascending  column.  For  good 
draught,  no  air  should  be  allowed  to  enter  the  flue  except  that 
which  has  passed  through  the  fire. 

574.  Ventilation  of  Apartments.— The  air  of  an  apart- 
ment, as  it  becomes  vitiated  by  respiration,  may  generally  be  re- 
moved, and  fresh  air  substituted,  by  taking  advantage  of  the  same 
inequality  of  weight  in  air-columns,  which  has  been  mentioned. 
If  opportunity  is  given  for  the  warm  impure  air  to  escape  from  the 
top  of  a  room,  and  for  external  air  to  take  its  place,  there  will  be 
a  constant  movement  through  the  room,  as  in  the  flue  of  a  chim- 
ney, though  at  a  slower  rate.  If  the  external  air  is  cold,  the 
weight  of  the  columns  differs  more,  and  therefore  the  ventilation 
is  more  easily  effected.  But  in  cold  weather,  the  air,  before  being 
admitted  to  the  room,  is  warmed  by  passing  through  the  air-cham- 
bers of  a  furnace.  When  there  is  a  chimney-flue  in  the  wall  of  a 
room,  with  a  current  of  hot  air  ascending  in  it,  the  ventilation  is 
best  accomplished  by  admitting  the  air  into  the  flue  at  the  upper 
part  of  the  room ;  since  it  will  then  be  removed  with  the  velocity 
of  the  hot-air  current. 

The  tendency  of  the  air  of  a  warm  room  to  pass  out  near  the 
top,  while  a  new  supply  enters  at  the  lower  part,  is  shown  by  hold- 
ing the  flame  of  a  candle  at  the  top,  and  then  at  the  bottom,  of  a 
door  which  is  opened  a  little  distance.  The  flame  bends  outward 
at  the  top  and  inward  at  the  bottom. 

The  impure  air  of  a  large  audience-room  is  sometimes  removed 
by  a  mechanical  contrivance,  as,  for  instance,  a  fan-wheel  placed 
above  an  opening  at  the  top,  and  driven  by  steam. 

The  ventilation  of  mines  is  accomplished  sometimes  by  a  fire 
built  under  a  shaft,  fresh  air  being  supplied  by  another  shaft,  and 
sometimes  by  a  fan-wheel  at  the  top  of  the  shaft.  If  there  happen 
to  be  two  shafts  which  open  to  the  surface  at  very  different  eleva- 
tions, ventilation  may  be  effected  by  the  inequality  of  temperature 
which  is  likely  to  exist  within  the  earth  and  above  it.  Let  MM 
(Fig.  296)  be  the  vertical  section  of  a  mine  through  two  shafts  A 
and  B,  which  open  at  different  heights  to  the  surface  of  the  earth. 
If  the  external  air  is  of  the  same  temperature  as  the  air  within 
the  earth,  then  the  column  A  in  the  longer  shaft  has  the  same 
weight  as  B  and  C  together,  measured  upward  to  the  same  level. 
In  that  case,  which  is  likely  to  occur  in  spring  and  fall,  there  is 
no  circulation  without  the  use  of  other  means.  But  in  summer 


340 


HEAT. 


V 


the  air  C  is  warmer  than  A  and  B ;  therefore  A  is  heavier  than 
B  +  G.     Hence  there  is  a  current  of  air  down  A  and  up  B.    In 
winter,  C  is  colder  than  air  within  the  earth ;  therefore  B  +  0 
are  together  heavier 
than    A,    and    the 
current  sets  in  the 
opposite     direction, 
down  B  and  up  A. 


575.  Sources 
of  Heat— TJie  sun, 
although  nearly  a 
hundred  millions  of 
miles  from  the  earth, 
is  the  source  of 
nearly  all  the  heat 
existing  at  its  sur- 
face. The  interior 

of  the  earth,  except  a  thickness  of  forty  or  fifty  miles  next  to  the 
surface,  is  believed  to  be  in  a  condition  of  heat  so  intense  that  all 
the  materials  composing  it  are  in  the  melted  state.  But  the  earth's 
crust  is  so  poor  a  conductor  that  only  an  insensible  fraction  of  all 
this  heat  reaches  the  surface. 

Mechanical  operations  are  usually  attended  by  a  development 
of  heat.  For  example,  if  a  broad  surface  of  iron  were  made  to  re- 
volve, rubbing  against  another  surface,  nearly  all  the  force  ex- 
pended in  overcoming  the  friction  would  appear  as  heat,  a  com- 
paratively small  part  being  conveyed  through  the  air  as  sound. 
The  cutting  tool  employed  in  turning  an  iron  shaft  has  been  known 
to  generate  heat  enough  to  raise  a  large  quantity  of  cold  water 
to  the  boiling  point,  and  to  keep  it  boiling  for  an  indefinite  time. 
It  is  a  fact  familiar  to  all,  that  violent  friction  of  bodies  against 
each  other  will  set  combustibles  on  fire.  The  axles  of  railroad 
cars  are  made  red-hot  if  not  duly  oiled ;  boats  are  set  on  fire  by 
the  rope  drawn  swiftly  over  the  edge  by  a  whale  after  he  is  har- 
pooned ;  a  stream  of  sparks  flies  from  the  emory  wheel  when  steel 
is  polished,  &c.  Condensation  and  percussion,  as  well  as  friction, 
and  all  sudden  applications  of  force,  cause  sensible  heat.  Indeed, 
wherever  the  full  equivalent  of  any  force  is  not  obtained  in  some 
other  form,  the  deficiency  may  be  detected  in  the  heat  which  is 
developed. 

Chemical  action  is  another  very  common  source  of  heat.  Com- 
bustion is  the  effect  of  violent  chemical  attraction  between  atoms 
of  different  natures,  when  both  light  and  heat  are  manifested.  If 
the  union  goes  on  slowly,  as  in  the  rusting  of  iron,  the  amount  of 


SOURCES    OF    HEAT.  341 

heat  is  the  same,  but  it  is  diffused  as  fast  as  developed.  The  molec- 
ular forces,  expended  in  most  cases  of  chemical  combination,  as 
measured  by  their  heating  effects,  are  enormously  great. 

The  warmth  produced  by  the  vital  processes  in  plants  and 
animals  is  supposed  by  many  physicists  to  be  caused  by  chemical 
action.  In  breathing  the  air,  some  of  its  oxygen  is  consumed, 
which  becomes  united  with  the  blood.  This  process  is  in  some 
respects  analogous  to  a  slow  combustion,  by  which  heat  is  evolved 
in  the  animal  system. 


PART    IX. 

T  . 


CHAPTER    I. 

MOTION  AND  INTENSITY  OF  LIGHT. 

576.  Definitions.  —  Light  is  supposed  to  consist  of  exceed- 
ingly minute  and  rapid  vibrations  in  a  medium  or  ether  which 
fills  space  ;  which  vibrations,  on  reaching  the  retina  of  the  eye, 
cause  vision,  as  the  vibrations  of  the  air  cause  hearing,  when  they 
impinge  on  the  tympanum  of  the  ear,  and  as  thermal  vibrations 
produce  a  sensation  of  warmth,  when  they  fall  on  the  skin. 

Bodies,  which  of  themselves  are  able  to  produce  vibrations  in 
the  ether  surrounding  them,  are  said  to  emit  light,  and  are  called 
self  -luminous,  or  simply  luminous  ;  those,  which  only  reflect  light, 
are  called  non-luminous.  Most  bodies  are  of  the  latter  class.  A 
ray  of  light  is  a  line,  along  which  light  is  propagated  ;  a  ~beam  is 
made  up  of  many  parallel  rays  ;  &  pencil  is  composed  of  rays  either 
diverging  or  converging  ;  and  is  not  unfrequently  applied  to  those 
which  are  parallel. 

A  substance,  through  which  light  is  transmitted,  is  called  a 
medium  ;  if  objects  are  clearly  seen  through  the  medium,  it  is 
called  transparent  ;  if  seen  faintly,  semi-transparent  ;  if  light  is 
discerned  through  a  medium,  but  not  the  objects  from  which  it 
comes,  it  is  called  translucent;  substances  which  transmit  no  light 
are  called  opaque. 

577.  Light  Moves  in  Straight  Lines.  —  So  long  as  the 
medium  continues  uniform,  the  line  of  each  ray  is  perfectly 
straight.      For  an  object  cannot  be  seen  through  a  bent  tube; 
and  if  three  disks  have  each  a  small  aperture  through  it,  a  ray 
cannot  pass  through  the  three,  except  when  they  are  exactly  in  a 
straight  line.     The  shadow  which  is  projected  through  space  from 
an  opaque  body  proves  the  same  thing;  for  the  edges  of  the 
shadow,  taken  in  the  direction  of  the  rays,  are  all  straight  lines. 


THE    VELOCITY    OF    LIGHT.  343 

From  every  point  of  a  luminous  surface  light  emanates  in  all 
possible  directions,  when  not  prevented  by  the  interposition  of 
an  opaque  body.  Thus,  a  candle  is  seen  by  night  at  the  distance 
of  one  or  two  miles ;  and  within  that  limit,  no  space  so  small  as 
the  pupil  of  the  eye  is  destitute  of  rays  from  the  candle.  A 
point  from  which  light  emanates  is  called  a  radiant.  If  light 
from  a  radiant  falls  perpendicularly  on  a  circular  disk,  the  pencil 
is  a  cone ;  if  on  a  square  disk,  it  is  a  square  pyramid,  &c.,  the 
illuminated  surface  in  each  case  being  the  base,  and  the  radiant 
the  vertex. 

578.  The  Velocity  of  Light. — It  has  been  ascertained  by 
several  independent  methods,  that  light  moves  at  the  rate  of  about 
192,500  miles  per  second. 

One  method  is  by  means  of  the  eclipses  of  Jupiter's  satellites. 
The  planet  Jupiter  is  attended  by  four  moons  which  revolve  about 
it  in  short  periods.  These  small  bodies  are  observed,  by  the  tele- 
scope, to  undergo  frequent  eclipses  by  falling  into  the  shadow 
which  the  planet  casts  in  a  direction  opposite  to  the  sun.  The 
exact  moment  when  the  satellite  passes  into  the  shadow,  or  comes 
out  of  it,  is  calculated  by  astronomers.  But  sometimes  the  earth 
and  Jupiter  are  on  the  same  side,  and  sometimes  on  opposite  sides 
of  the  sun ;  consequently,  the  earth  is,  in  the  former  case,  the 
whole  diameter  of  its  orbit,  or  about  one  hundred  and  ninety 
millions  of  miles  nearer  to  Jupiter  than  in  the  latter.  Now  it  is 
found  by  observation,  that  an  eclipse  of  one  of  the  satellites  is 
seen  about  sixteen  minutes  and  a  half  sooner  when  the  earth  is 
nearest  to  Jupiter,  than  when  it  is  most  remote  from  it,  and  con- 
sequently, the  light  must  occupy  this  time  in  passing  through  the 
diameter  of  the  earth's  orbit,  and  must  therefore  travel  at  the  rate 
of  about  192,000  miles  per  second. 

Another  method  of  estimating  the  velocity  of  light,  wholly 
independent  of  the  preceding,  is  derived  from  what  is  called  the 
aberration  of  the  fixed  stars.  The  apparent  place  of  a  fixed  star 
is  altered  by  the  motion  of  its  light  being  combined  with  the  mo- 
tion of  the  earth  in  its  orbit.  The  place  of  a  luminous  object  is 
determined  by  the  direction  in  which  its  light  meets  the  eye.  But 
the  direction  of  the  impulse  of  light  on  the  eye  is  modified  by  the 
motion  of  the  observer  himself,  and  the  object  appears  forward  of 
its  true  place.  The  stars,  for  this  reason,  appear  slightly  displaced 
in  the  direction  in  which  the  earth  is  moving ;  and  the  velocity 
of  the  earth  being  known,  that  of  light  may  be  computed  in  the 
same  manner  as  we  determine  one  component,  when  the  angles 
and  the  other  component  are  known. 

The  velocity  of  light  has  been  determined  also  by  direct  ex- 


344  LIGHT. 

pertinent,  in  a  manner  somewhat  analogous  to  that  employed  by 
Wheatstone  for  ascertaining  the  velocity  of  electricity. 

579.  Loss  of  Intensity  by  Distance. — The  intensity  of 
ligkt  varies  inversely  as  the  square  of  the  distance.    In  Fig.  297, 
suppose  light  to  radi- 
ate from  S,  through  FlG- 

the  rectangle  A  C9 
and  fall  on  E  G,  paral- 
lel to  A  C.  As  8  A  E, 
S  B  F,  &c.,  are 
straight  lines,  the  tri- 
angles, SAB,  SE F, 
are  similar,  as  also  the 
rectangles,  A  C\  E  G  ;  therefore,  A  C :  E  G  : :  A  B> :  EF2 : :  SA* : 
S  E\  But  the  same  quantity  of  light,  being  diffused  over  A  C  and 
E  G,  will  be  more  intense,  as  the  surface  is  smaller.  Hence,  the 
intensity  of  light  at  E :  intensity  at  A  : :  A  C :  E  G  : :  S  A2 :  S  E1, 
which  proves  the  proposition.  This  demonstration  is  applicable 
to  every  kind  of  emanation  in  straight  lines  from  a  point. 

580.  Brightness  the  Same  at  all  Distances. — The  Irujlit- 
ness  of  an  object  is  the  quantity  of  light  which  it  sheds,  as  com- 
pared with  the  apparent  area  from  which  it  comes.    Now  the  quan- 
tity (or  intensity),  as  has  just  been  shown,  varies  inversely  as  the 
square  of  the  distance.     The  apparent  area  of  a  given  surface  also 
diminishes  in  the  same  ratio,  as  we  recede  from  it.     Hence,  the 
brightness  is  constant.    For  illustration,  if  we  remove  to  three 
times  the  distance  from   a  luminous   body,  we  receive  into   the 
eye  nine  times  less  light,  but  the  body  also  appears  nine  times 
smaller,  so  that  the  relation  of  light  to  apparent  area  remains  the 
same. 

581.  Loss  of  Intensity  by  Absorption. — In  a  uniform 
medium,  while  the  distance  increases  arithmetically,  the  intensity 
diminishes  geometrically.     Imagine  the  medium  to  be  divided  by 
parallel  planes  into  strata  of  equal  thickness;   and  suppose  the 

first  stratum  to  diminish  the  intensity  by  -  of  the  whole.    Then 
the  intensity  of  the  light  which  reaches  the  second  stratum  is 

1 =  -     — .     But  on  account  of  the  uniformity  of  the  me- 

n          n  J 

dium,  every  stratum  produces  the  same  effect,  that  is,  it  transmits 

to  the  next, of  that  which  falls  upon  it.    Therefore,  - 

n  n 


SHADOWS.  345 

of  —    —  ,  or  ^-  —  5—  -  ,  leaves  the  second  stratum,  -~~a  '-,  the  third, 

and  so  on,  in  a  geometrical  series.  For  example,  if  a  piece  of 
colored  glass  is  If  inch  thick,  and  each  quarter  of  an  inch  ab- 
sorbs |-  of  the  light  which  falls  upon  it,  then  about  one-hun- 
dredth of  what  enters  the  first  surface  will  escape  from  the  last. 


For        =  .01  nearly. 

582.  Photometers.  —  These  are  instruments  designed  for  the 
measurement  of  the  relative  intensities  of  light.    We  cannot  de- 
termine by  the  eye  alone  how  many  times  more  intense  one  light 
is  than  another,  though  we  can  judge  with  tolerable  accuracy 
when   two   surfaces   are   equally  illuminated.     Photometers   are, 
therefore,  generally  constructed  on  the  plan  of  determining  the 
ratio  of  intensities  of  two  lights,  by  means  of  our  ability  to  decide 
when  they  illuminate  two  surfaces  equally.    It  is  sufficient  to 
mention  Rumford's  method  by  shadows.    Let  the  two  unequal 
lights  be  so  placed  that  the  two  shadows  of  an  opaque  body  cast 
by  them  shall  fall  side  by  side  on  a  white  screen.    If  one  shadow 
appears  more  luminous  than  the  other,  remove  to  a  greater  dis- 
tance the  light  which  illuminates  it  (or  bring  the  other  nearer), 
until  the  shadows  appear  of  the  same  degree  of  illumination. 
Then  measure  the  distances  from  the  lights  to  the  screen,  and  the 
intensities  of  the  lights  will  be  directly  as  the  squares  of  the  dis- 
tances.   For  the  light  at  the  greater  distance,  since  it  illuminates 
the  screen  equally  with  the  other,  must  gain  as  much  by  intensity 
as  it  loses  by  distance  ;  that  is,  in  the  ratio  of  the  square  of  the 
distance. 

583.  Shadows.  —  When  a  luminous  body  shines  on  one  which 
is  opaque,  the  space  beyond  the  latter,  from  which  the  light  is 
excluded,  is  called  a  shadow.     The  same  word,  as  commonly  used, 
denotes  only  the  section  of  a  shadow  made  by  a  surface  which 
crosses  it.     Shadows  are  either  total  or  partial.    If  tangents  are 
drawn  on  all  the  corresponding  sides  of  the  two  bodies,  the  space 
inclosed  by  them  beyond  the  opaque  body  is  the  total  shadow;  if 
other  tangents  are  drawn,  crossing  each  other  between  the  bodies, 
the  space  between  the  total  shadow  and  the  latter  system  of  tan- 
gents is  the  partial  shadow,  or  penumbra.    In  case  the  bodies  are 
spheres,  as  in  Fig.  298,  the  total  shadow  will  be  a  cylinder,  or  con- 
ical frustum,  each  of  infinite  length,  or  a  complete  cone,  according 
to  the  relative  size  of  the  spheres.     But,  in  every  case,  the  penum- 
bra and  inclosed  total  shadow  will  form  an  increasing  frustum. 
It  is  obvious  that  the  shade  of  the  penumbra  grows  gradually 
deeper  from  the  outer  surface  to  the  total  shadow  within  it. 


346 


LIGHT. 


Every  shadow  cast  by  the  sun  has  a  penumbra  bordering  it, 
which  gives  to  the  shadow  an  ill-defined  edge ;  and  the  more  re- 


Fia.  298. 


mote  the  sectional  shadow  is  from  the  opaque  body  which  casts  it, 
the  broader  will  be  the  partial  shadow  on  the  edge. 


CHAPTER    II. 

EEFLECTION    OF    LIGHT. 

584.  Radiant  and  Specular  Reflection.— Light  is  said  to 
be  reflected  when,  on  meeting  a  surface,  it  is  turned  back  into  the 
same  medium.  In  ordinary  cases  of  reflection,  the  light  is  diffused 
in  all  directions,  and  it  is  by  means  of  the  light  thus  scattered 
from  a  body  that  it  becomes  visible,  when  it  sheds  no  light  of  its 
own.  This  is  called  radiant  reflection.  It  is  produced  by  unpol- 
ished surfaces.  But  when  a  surface  is  highly  polished,  a  beam  of 
light  falling  on  it  is  reflected  in  some  particular  direction ;  and, 
if  the  eye  is  placed  in  this  reflected  beam,  it  is  not  the  reflecting 
surface  which  is  seen,  but  the  original  object,  apparently  in  a  new 
position.  This  is  called  specular  reflection.  It  is,  however,  gene- 
rally accompanied  by  some  degree  of  radiant  reflection,  since  the 
reflector  itself  is  commonly  visible  in  all  directions.  Ordinary 
mirrors  are  not  suitable  for  accurate  experiments  on  reflection, 
because  light  is  modified  by  the  glass  through  which  it  passes. 
The  speculum  is  therefore  used,  which  is  a  reflector  made  of  solid 
metal,  and  accurately  ground  to  any  required  form,  either  plane, 
convex,  or  concave.  The  word  mirror  is,  however,  much  used  in 
optics  for  every  kind  of  reflector. 


THE    LAW    OF    REFLECTION 


347 


FIG.  299. 


Optical  experiments  are  usually  performed  on  a  beam  of  light 
admitted  through  an  aperture  into  a  darkened  room;  the  direction 
of  the  beam  being  regulated  by  an  adjustable  mirror  placed  out- 
side. An  instrument  consisting  of  a  plane  speculum  moved  by  a 
clock,  in  such  a  manner  that  the  reflected  sunbeam  shall  remain 
stationary  at  all  hours  of  the  day,  is  called  a  heliostat. 

585.  The  Law  of  Reflection. — When  a  ray  of  light  is  inci- 
dent on  a  mirror,  the  angle  between  it  and  a  perpendicular  to  the 
surface  at  the  point  of  incidence,  is  called  the  angle  of  incidence; 
and  the  angle  between  the  reflected  ray  and  the  same  perpendicu- 
lar, is  called  the  angle  of  reflection.  The  law  of  reflection  found 
to  be  universally  true  is  the  following : 

Tlie  angles  of  incidence  and  reflection  are  on  opposite  sides  of 
the  perpendicular,  and  are  equal  to  each  other. 

This  is  well  shown  by  attaching  a  small  mirror  to  the  centre 
of  a  graduated  semicircle  perpendicular  to  its  plane.  Let  M  D  N 
(Fig.  299)  be  the  semicircle,  graduated  from  D  both  ways  to  M 
and  N,  and  mounted  so  that  it  can 
be  revolved  on  its  centre,  and 
clamped  in  any  position.  Let  the 
small  mirror  be  at  C,  with  its  plane 
perpendicular  to  C D\  then  a  ray 
from  the  heliostat,  as  A  (7,  passing 
the  edge  at  a  particular  degree, 
will  be  seen  after  reflection  to  pass 
the  corresponding  degree  in  the 
other  quadrant.  By  revolving  the 
semicircle,  any  angle  of  incidence 
may  be  tried,  and  the  two  rays  are 
always  found  to  be  in  the  same 
plane  with  CD,  and  equally  in- 
clined to  it. 

As  the  mirror  revolves,  the  re- 
flected ray  revolves  tivice  as  fast. 

For  A  CD  is  increased  or  diminished  by  the  angle  through 
which  the  mirror  turns;  therefore  D  C  B  is  also  increased  or 
diminished  by  the  same;  hence  ACS,  the  angle  between  the  two 
rays,  is  increased  or  diminished  by  the  sum  of  both,  or  twice  the 
same  angle. 

It  follows  from  the  law  of  reflection,  that  a  ray  which  falls  on 
a  mirror  perpendicularly,  retraces  its  own  path  after  reflection.  It 
is  obvious,  also,  that  the  complements  of  the  angles  of  incidence 
and  reflection  are  equal,  i.  e.  A  C  M  —  B  C  N.  The  law  of  reflec- 
tion is  applicable  to  curved  as  well  as  to  plane  mirrors ;  the  radius 


348  LIGHT. 

of  curvature  at  any  point  being  the  perpendicular  with  which  the 
incident  and  reflected  rays  make  equal  angles. 

Radiant  reflection  forms  no  exception  to  the  foregoing  law, 
though  the  incident  rays  are  in  one  and  the  same  direction,  and 
the  reflected  rays  are  scattered  every  way.  For  the  minute  cavi- 
ties and  prominences  which  constitute  the  roughness  of  the  gene- 
ral surface  are  bounded  by  small  surfaces  lying  at  all  inclinations ; 
and  each  one  reflecting  the  rays  which  meet  it  in  accordance  with 
the  law,  those  rays  are  necessarily  thrown  off  in  all  possible 
directions. 

586.  Inclination  of  Rays  to  each  other  not  altered  by 
the  Plane  Mirror. — 

1.  Rays  which  diverge  before  reflection,  diverge  at  the  same 
angle  after  reflection. 

Let  M N  (Fig.  300)  be  a  plane  mirror, 
and  A  B,  A  C,  any  two  rays  of  light  fall- 
ing upon  it  from  the  radiant  A,  and  re- 
flected in  the  lines  B  E,  C  G.  Draw  the 
perpendicular  A  P,  and  produce  it  indefi- 
nitely, as  to  F,  behind  the  mirror;  also 
produce  the  reflected  rays  back  of  the 
mirror.  Let  Q  R  be  perpendicular  to  the 
mirror  at  the  point  B\  it  is  therefore 
parallel  to  A  F,  and  the  plane  passing 
through  A  F  and  Q  R,  is  that  which  in- 
cludes the  ray  A  B.  B  E.  Therefore, 

E  B,  when  produced  back  of  the  mirror,  intersects  A  P  produced. 
Let  F  be  the  point  of  intersection.  B  A  F  =  A  B  ft  and  A  FB 
—  EBQ\  }miABQ  =  EBQ  (Art.  585);  .\BAF-AFB, 
and  A  B—  F  B.  If  P  and  B  be  joined,  P  B  being  in  the  plane 
M  Ni&  perpendicular  to  A  F,  and  therefore  bisects  it.  Hence,  the 
reflected  ray  meets  the  perpendicular  A  F  as  far  behind  the  mir- 
ror, as  the  incident  ray  does  in  front.  In  the  same  way  it  may  be 
proved  that  A  C  —  0  F,  and  that  C  G,  when  produced  back  of  the' 
mirror,  meets  A  F  at  the  same  point  F. 

Now,  since  the  triangles  A  C  B  and  F  C  B,  have  their  sides 
respectively  equal,  their  angles  are  equal  also ;  hence  BAG  — 
B  F  G.  Therefore  any  two  rays  diverge  at  the  same  angle  after 
reflection  as  they  did  before  reflection. 

Since  the  reflected  rays  seem  to  emanate  from  F,  that  point  is 
called  the  apparent  radiant ;  A  is  the  real  radiant. 

2.  Rays  which  converge  before  reflection,  converge  at  the  same 
angle  after  reflection.  Let  E  B,  G  C,  be  incident  rays  converging 
toward  F,  and  let  B  A,  C  A,  be  the  reflected  rays.  It  may  be 


CONCAVE    MIRROR.  349 

proved  as  before,  that  A  and  .Fare  in  the  same  perpendicular,  A  F, 
and  equidistant  from  P,  and  that  E  F  G  =  B  A  C. 

The  point  F,  to  which  the  incident  rays  were  converging,  is 
called  the  virtual  focus  ;  A  is  the  real  focus. 

3.  Eays  which  are  parallel  before  reflection  are  parallel  after 
reflection. 

It  has  been  proved  in  case  1,  that  F,  the  intersection  of  the 
reflected  rays,  is  as  far  behind  the  mirror,  as  A,  the  intersection  of 
incident  rays,  is  before  it.  Now,  if  the  incident  rays  are  parallel, 
A  is  at  an  infinite  distance  from  the  mirror.  Therefore  F  is  at 
an  infinite  distance  behind  it,  and  the  reflected  rays  are  parallel. 

In  all  cases,  therefore,  rays  reflected  by  a  plane  mirror  retain 
the  same  inclination  to  each  other  which  they  had  before  reflection. 

587.  Spherical  Mirrors. — A  spherical  mirror  is  one  which 
forms  a  part  of  the  surface  of  a  sphere,  and  is  either  convex  or 
concave.     The  axis  of  such  a  mirror  is  that  radius  of  the  sphere 
which  passes  through  the  middle  of  the  mirror.    In  the  practical 
use  of  spherical  mirrors,  it  is  found  that  the  light  must  strike  the 
surface  very  nearly  at  right  angles ;  hence,  in  the  following  state- 
ments, the  mirror  is  supposed  to  be  a  very  small  part  of  the  whole 
spherical  surface,  and  the  rays  nearly  coincident  with  the  axis. 

It  is  sufficient  to  trace  the  course  of  the  rays  on  one  side  of  the 
axis,  since,  on  account  of  the  symmetry  of  the  mirror  around  the 
axis,  the  same  effect  is  produced  on  every  side. 

588.  Converging  Effect  of  a  Concave  Mirror. — 

1.  Parallel  rays  are  converged  to  the  middle  point  between  the 
centre  and  surface,  which  is  therefore  called  the  focus  of  parallel  rays, 
or  the  principal  focus.  Let  R  A,  L  E  (Fig.  301),  be  parallel  rays 
incident  upon  the  concave  mirror  A  B,  whose  centre  of  concavity  is 

FIG.  301. 


C.  The  ray  L  E,  passing  through  C,  and  therefore  perpendicular 
to  the  mirror  at  E,  is  reflected  directly  back.  Join  C  A,  and  make 
C  A  F=  R  A  Cj  then  R  A  is  reflected  in  the  line  A  F,  and  the 
two  reflected  rays  meet  at  F.  R  A  C=  A  C  Ft  .:  A  CF=  FA  C, 
and  A  F—  C F\  and  as  A  and  E  are  very  near  together,  E  F  — 
F  C\  that  is,  the  focus  of  parallel  rays  is  at  the  middle  point  be- 
tween C  and  E. 


350  LIGHT. 


2.  Diverging  rays,  falling  on  a  given  concave  mirror,  are  re- 
flected converging,  parallel,  or  less  diverging,  according  to  the 
degree  of  divergency  in  the  original  pencil.  Let  C  (Fig.  302)  be 
the  centre  of  concavity,  and  F  the  focus  of  parallel  rays.  Then, 


FIG.  303. 


rays  diverging  from  any  point,  A,  beyond  C,  will  be  converged  to 
some  point,  a,  between  C  and  F,  since  the  angles  of  incidence  and 
reflection  are  less  than  those  for  parallel  rays.  Eays  diverging 
from  C  are  reflected  back  to  (7;  those  from  points  between  C  and 
F,  as  a,  are  converged  to  points  beyond  C9  as  A ;  those  diverging 
from  .F  become  parallel;  and  those  from  points  between  F  and  the 
mirror,  as  D,  diverge  after  reflection,  but  at  a  less  angle  than  be- 
fore, and  seem  to  flow  from  A'.  To  prove,  in  the  last  case,  that 
the  angle  of  divergence,  A',  after  reflection,  is  less  than  the  angle 
D,  the  divergence  before  reflection,  observe  that  the  angle  A'  is 
less  than  the  exterior  angle  H B  C\  but  H B  C=D  B  C  (Art. 
585),  which  is  less  than  D  B  R,  which  is  equal  to  A' D  B\  much 
more,  then,  is  A'  less  than  A'  D  B. 

3.  Converging  rays  are  made  to  converge  more.  The  rays  H  B, 
A  E,  converging  to  A',  are  reflected  to  D,  nearer  the  mirror  than 
F  is.  And  it  has  been  shown  that  the  angle  D  is  larger  than  A ', 
hence  the  convergency  is  increased. 

From  the  three  foregoing  cases,  it  appears  that  the  concave 
mirror  always  tends  to  produce  convergency ;  since,  when  it  does 
not  actually  produce  it,  it  diminishes  divergency. 

589.  Conjugate  Foci. — When  light  radiates  from  A,  it  is 
reflected  to  #;  when  it  radiates  from  a,  it  meets  at  A.  Any  two 
such  interchangeable  points  are  called  conjugate  foci.  If  the  radius 
of  the  mirror  and  the  distance  of  one  focus  from  the  mirror  are 
given,  the  distance  of  its  conjugate  focus  may  be  determined. 
Let  the  radius  =  r ;  the  distance  A  E  =  m  •  and  a  E  =  n.  As 
the  angle  A  B  a  is  bisected  by  B  C,  A  B  :  a  B  : :  A  C :  a  C\  that 
is,  since  B  E  is  very  small,  A  E :  a  E : :  A  C:  a  C,  or,  mini: 
m  —  r  :  r  —  n. 

n  r  m  r 

/.    m  =  - :  and  n  =  s . 

2  n  —  r'  2m  —  r 

If  A  is  not  on  the  axis  of  the  mirror,  as  in  Fig.  303,  let  a  line 


CONVEX    MIRROR. 


351 


FIG.  803. 


be  drawn  through  A  and  C,  meeting  the  mirror  in  E;  this  is  called 
a  secondary  axis,  and  the  light  radiating  from  A  will  be  reflected 
to  a  on  the  same  secondary  axis, 
for  A  E  is  perpendicular  to  the 
mirror,  and  will  be  reflected  di- 
rectly back ;  and  if  A  E  and  C  E 
are  given,  a  E  may  be  found  as 
before. 

590.  Diverging  Effect  of  a  Convex  Mirror. — 

1.  Parallel  rays  are  reflected  diverging  from  the  middle  point 
between  the  centre  and  surface.  Let  C  (Fig.  304)  be  the  centre 
of  convexity  of  the  mirror  M N,  and  draw  the  radii,  G M,  CD, 

FIG.  304. 


producing  them  in  front  of  the  mirror ;  these  are  perpendicular  to 
the  surface.  The  ray  R  D  will  be  reflected  back ;  A  M  will  be 
reflected  in  M  B,  making  B  M  E  —  A  ME.  Produce  the  re- 
flected ray  back  of  the  mirror,  and  it  will  meet  the  axis  in  F,  mid- 
way from  C  to  Z>;  for  FCM=A  M  E,  and  F  M  C=  B  M  E\ 
therefore  the  triangle  F  C  Mis  isosceles,  and  C  F  —  F  M,  and  as 
M  is  very  near  D,  C  F—  F  D.  Hence  the  rays,  after  reflection, 
diverge  as  if  they  radiated  from  a  point  in  the  middle  of  C  D} 
which  is  the  apparent  radiant. 

2.  Diverging  rays  have  their  divergency  increased.    Let  A  D, 
A  M(Fig.  305),  be  the  diverging  rays;  D  A,  MB,  the  reflected 

FIG.  305. 


rays ;  these  when  produced  meet  at  F,  which  is  the  apparent  radi- 
ant. M  A  Fis  the  divergency  of  the  incident  rays,  and  A  F  B 
of  the  reflected  rays.  Now  the  exterior  angle,  A  F  B,  is  greater 
than  C  M  F,  or  B  M  E,  or  A  M  E.  But  A  M  E,  being  exterior, 
is  greater  than  M  A  F-,  much  more,  then,  is  A  F  B  greater  than 
MA  F. 

3.  Convergent  rays  are  at  least  rendered  less  convergent,  and 


352 


LIGHT. 


may  become  parallel  or  divergent,  according  to  the  degree  of  pre- 
vious convergency.  The  two  first  effects  are  shown  by  Figs.  304 
and  305,  reversing  the  order  of  the  rays.  And  it  is  easy  to  per- 
ceive that  rays  converging  to  (7,  will  diverge  from  C  after  reflec- 
tion ;  if  to  a  point  more  distant  than  O,  they  will  diverge  afterward 
from  a  point  between  C  and  F  (Fig.  304),  and  vice  versa. 

The  general  effect,  therefore,  of  a  convex  mirror,  is  to  produce 
divergency. 

A  and  F  (Fig.  305)  are  called  conjugate  foci,  being  inter- 
changeable points ;  for  rays  from  A  move  after  reflection  as  though 
from  F,  and  rays  converging  to  F  are  by  reflection  converged  to 
A.  Conjugate  foci,  in  the  case  of  the  convex  mirror,  are  in  the 
same  axis  either  principal  or  secondary,  as  they  are  in  the  concave 
mirror,  and  for  the  same  reason,  viz.,  that  every  axis  is  perpendic- 
ular to  the  surface. 

591.  Images  by  Reflection.— An  optical  image  consists  of 
a  collection  of  focal  points,  from  which  light  either  really  or  appa- 
rently radiates.     "When  rays  are  converged  to  a  focus  they  do  not 
stop,  but  cross,  and  diverge  again,  as  if  originally  emanating  from 
the  focal  point.     A  collection  of  such  points,  arranged  in  order, 
constitutes  a  real  image.    When  rays  are  reflected  diverging,  they 
proceed  as  though  they  emanated  from  a  point  behind  the  mirror. 
A  collection  of  such  imaginary  radiants  forms  an  apparent  or  vir- 
tual image.    The  images  formed  by  plane  and  convex  mirrors  are 
always  apparent;   those  formed  by  concave  mirrors  may  be  of 
either  kind. 

592.  Images  by  a  Plane  Mirror. — When  an  object  is  before 
a  plane  mirror,  its  image  is  at  the  same  distance  behind  it,  of  the 
same  magnitude,  and  equally  inclined  to  it.    Let  M  N  (Fig.  306) 
be  a  plane  mirror,  and  A  B  an  ob- 
ject before  it,  and  let  the  position  '  FlG-  306. 

of  the  object  be  such  that  the  re- 
flected rays  may  enter  the  eye 
placed  at  H.  From  A  and  B  let 
fill  upon  the  plane  of  the  mirror 
t'.ie  perpendiculars  A  E,  B  G,  and 
produce  them,  making  Ea  —  A  E, 
and  G  b  =  B  G.  Now,  since  the 
rays  from  A  will,  after  reflection, 

radiate  as  if  from  a  (Art.  586),  and  those  from  B,  as  if  from  5,  and 
the  same  of  all  other  points,  therefore  the  image  and  object  are 
equally  distant  from  the  mirror.  A  C,  a  c,  parallel  to  the  mirror, 
are  equal ;  as  B  G  =  b  G,  and  A  E  =  a  E,  therefore,  by  subtrac- 
tion, B  C  =  b  c ;  also  the  right  angles  at  C  and  c  are  equal.  There- 


IMAGES    BY    PLANE    MIRRORS.  353 

fore  A  B  =  a  I,  and  B  A  C  =t>  ac;  that  is,  the  object  and  image 
are  of  equal  size,  and  equally  inclined  to  the  mirror. 

It  appears  from  the  demonstration,  that  the  object  and  its 
image  are  comprehended  between  the  same  perpendiculars  to  the 
plane  of  the  mirror. 

The  object  and  image  obviously  have  to  each  other  twice  the 
inclination  that  each  has  to  the  mirror.  Hence,  in  a  mirror  in- 
clined 45°  to  the  horizon,  a  horizontal  surface  appears  vertical, 
and  one  which  is  vertical  appears  horizontal. 

593.  Symmetry  of  Object  and  Image. — All  the  three  di- 
mensions of  the  object  and  image  are  respectively  equal,  as  shown 
above,  but  one  of  them  is  inverted  in  position,  namely,  that  dimen- 
sion which  is  perpendicular  to  the  mirror.     Hence,  a  person  and 
his  image  face  in  opposite  directions ;  and  trees  seen  in  a  lake  have 
their  tops  downward.    Those  dimensions  which  are  parallel  to  the 
mirror  are  not  inverted.    In  consequence  of  the  inversion  of  one 
dimension  alone,  the  object  and  its  image  are  not  similar,  but 
symmetrical  forms ;  and  one  could  not  coincide  with  the  other  if 
brought  to  occupy  the  same  space.    The  image  of  a  right  hand  is 
a  left  hand,  and  all  relations  of  right  and  left  are  reversed.    It  is 
for  this  reason  that  a  printed  page,  seen  in  a  mirror,  is  like  the 
type  with  which  it  was  printed. 

594.  The  Length  of  Mirror  Requisite  for  Seeing  an 
Object. — If  an  object  is  parallel  to  a  mirror,  the  length  of  mirror 
occupied  by  the  image  is  to  the  length  of  the  object  as  the  reflected 
ray  to  the  sum  of  the  incident  and  reflected 

rays.  Let  A  B  (Fig.  307)  be  the  length  of  the 
object,  CD  that  of  the  image,  and  F  G  that  of 
the  space  occupied  on  the  mirror ;  then,  by 
similar  triangles,  F G  :  C 'D  \\EF\E C.  But 
CD  =  AB,  and  CF=AF;  .-.FGiABi: 
EF  :  A  F  +  F  E.  If  the  eye  is  brought 
nearer  the  mirror,  the  space  on  the  mirror  oc- 
cupied by  the  image  is  diminished,  because  E  F 
has  to  A  F  +  F  E  a  less  ratio  than  before.  The  same  effect  is  pro- 
duced by  removing  the  object  further  from  the  mirror.  The  length 
of  mirror  necessary  for  a  person  to  see  himself  is  equal  to  half  his 
height,  because  in  that  case,  EF\  A  F  +  F  E : :  I  :  2,  which  ratio 
will  not  be  altered  by  change  of  distance. 

595.  Displacement  of  Image  by  Two  Reflections. — If  an 

image  is  seen  by  light  reflected  from  two  mirrors  in  a  plane  per- 
pendicular to  their  common  section,  its  angular  deviation  from 
23 


354  LIGHT. 

the  object  is  equal  to  twice  the  'inclination  of  the  mirrors.  Let 
AB  CD  (Fig.  308)  be  two  plane  mirrors  inclined  at  the  angle 
AGO.  If  an  eye  at  H  sees  the 
star  Sin  the  direction  0,  the  an-  s/  ] 


%GBD.    In  like  manner,  BDO      A 


-HBD=2BDC-2GBD 


This  principle  is  employed  in 
the  construction  of  Hartley's  quad- 
rant, and  the  sextant,  used  at  sea 
for  measuring  angular  distances. 
The  angles  measured  are  twice  as 
great  as  the  arc  passed  over  by 
the  index  which  carries  the  re-  G 

volving  mirror ;    hence,  in  the 

quadrant,  an  arc  of  45°  is  graduated  into  90° ;  and,  in  the  sextant, 
an  arc  of  60°  is  graduated  into  120°. 

596.  Multiplied  Images  by  Two  Mirrors. — 

1.  Parallel  Mirrors.  The  series  of  images  is  infinite  in  num- 
ber, and  arranged  in  a  straight  line,  perpendicular  to  the  mirrors. 
The  object  E  between  the  parallel  mirrors,  A  B,  CD  (Fig.  309), 


FIG.  309. 

A          C 

z*    &      JF|  x 

—  V/~  "W" 

Cr          J<            & 

-V  V- 

M 

1 

has  an  image  at  F,  as  far  behind  A  B  as  E  is  in  front  of  it,  and 
between  the  same  perpendiculars.  The  rays  reflected  by  A  B 
diverge,  as  though  they  emanated  from  F',  hence,  F  may  be  re- 
garded as  an  object  before  CD,  whose  image  is  at  F',  as  far  behind 
it.  Again,  F'  may  be  considered  as  an  object  before  A  B,  and  so 
on  indefinitely.  Another  series  exists  in  the  same  line,  by  begin- 
ning with  G,  the  first  image  behind  CD.  As  light  is  absorbed 
and  scattered  by  each  reflection,  these  images  grow  fainter,  and  at 
length  disappear.  Articles  of  jewelry  are  sometimes  apparently 
multiplied  and  extended  over  a  large  surface,  by  lining  the  cases 
with  parallel  mirrors. 


MULTIPLIED    IMAGES    BY    TWO    MIRRORS.      355 


The  multiplied  images  of  a  small  bright  object,  sometimes 
seen  in  a  looking-glass,  are  produced  by  repeated  reflections  be- 
tween the  front  and  the  silvered  covering  on  the  back  side.  At 
each  internal  impact  on  the  first  surface  some  light  escapes,  and 
shows  us  an  image,  while  another  portion  is  reflected  to  the  back, 
and  thence  forward  again.  The  image  of  a  lamp  viewed  very 
obliquely  in  a  mirror  is  sometimes  repeated  eight  or  ten  times  ; 
and  a  planet,  or  bright  star,  when  seen  in  a  looking-glass,  will 
be  accompanied  by  three  or  four  faint  images,  caused  in  the 
same  way. 

2.  Inclined  Mirrors.  In  this  case,  the  images  are  limited  in 
number,  and  arranged  in  the  circumference  of  a  circle,  whose 
centre  is  in  the  line  of  common  section  of  the  planes  of  the  mir- 
rors, and  whose  radius  is  the  distance  of  the  object  from  that  line. 
Let  A  B,  A  C  (Fig,  310)  be  the  mirrors,  and  E  the  object.  Draw 
E  G  perpendicular  to  A  B, 
and  make  E  F  =  F  G,  then 
will  G  be  the  first  image  :  in 
the  same  way,  find  /,  the  im- 
age of  G  by  A  C'y  K,  the 
image  of  /;  and  F,  that  of 
K.  Then  begin  with  the 
mirror  A  C,  and  find,  as  be- 
fore, M,  Of  P,  Q,  the  succes- 
sive images  by  the  two  mir- 
rors. No  image  of  V  or  Q 
can  be  formed,  because  they 
are  behind  both  mirrors.  All 
these  images  are  in  the  cir- 
cumference of  a  circle,  whose 
radius  is  E  A ;  for  EF,F  A, 

and  angle  at  F,  are  respectively  equal  to  G  F,  F  A,  and  angle  at 
F;  /.  E  A  =  G  A  ;  and  in  the  same  way  it  may  be  proved,  that 
E  A  =  A  My  A  I,  &c.  If  the  edges  at  A  be  separated,  making 
the  inclination  of  the  mirrors  less  and  less,  the  number  of  images 
will  increase,  and  the  circumference  will-  approach  a  straight  line, 
so  that  ultimately  we  shall  have  the  case  described  in  (1),  in  which 
the  mirrors  are  parallel. 

597.  Path  of  the  Pencil  by  which  each  Image  is  Seen.— 

Fig.  311  will  assist  to  understand  how  each  image  is  seen  by  a 
pencil  of  light  which  passes  back  and  forth  between  the  mirrors, 
until  it  reaches  the  eye.  If  the  eye  is  at  0,  and  the  object  at  Q, 
and  its  images  at  A,  B,  C,  D,  each  image  is  of  course  seen  by  a 
pencil  which  comes  from  the  mirror  to  the  eye,  as  if  it  originated 


356 


LIGHT. 


FIG.  311. 


in  that  image.  Therefore,  draw  a  line  from  any  image  as  D,  to 
the  eye,  and  from  its  intersection  with  the  mirror  draw  a  line 
to  the  preceding  image  ; 
from  the  intersection  of 
that  line  with  the  other 
mirror,  a  line  to  the  im- 
age next  preceding,  and 
so  on  back  to  Q;  the 
whole  path  of  the  pen- 
cil will  then  be  traced. 
Thus,  A  0  being  joined, 
and  Q  a  drawn  to  the  in- 
tersection a,  the  image  A 
is  seen  by  the  ray  Q  a, 
a  0.  In  like  manner,  B  is 
seen  by  Q  1),  It  c,  c  0 ;  (7, 
by  Qd,  de,ef,fO;  and 
A  hy  Q  g,  g  h,  h  i,  i  k, 
JcO. 

598.  The  Kaleidoscope. — This  instrument,  when  carefully 
constructed,  beautifully  exhibits  the  phenomenon  of  multiplied 
reflection  by  inclined  mirrors.     It  consists  of  a  tube  containing 
two  long,  narrow,  metallic  mirrors,  inclined  at  a  suitable  angle ; 
and  is  used  by  placing  the  objects  (fragments  of  colored  glass, 
etc.)  at  one  end,  and  applying  the  eye  to  the  other.    In  order  that 
there  may  be  perfect  symmetry  in  the  figure  made  up  of  the  ob- 
jects and  their  successive  images,  the  angle  of  the  mirrors  should 
be  of  such  size,  that  it  can  be  exactly  contained  an  even  number 
of  times  in  360q.    The  best  inclination  is  30°  ;  and  the  field  of 
view  is  then  composed  of  12  sectors.    It  is  also  essential,  that  the 
small  objects  forming  the  picture,  should  lie  at  the  least  possible 
distance  beyond  the  mirrors.     To  insert  three  mirrors  instead  of 
two,  as  is  often  done,  only  serves  to  confuse  the  picture,  and  mar 
its  beauty. 

599.  Images  by  the  Concave  Mirror.— The  concave  mir- 
ror forms  various  images,  either  real  or  apparent,  either  greater  or 
less  than  the  object,  either  erect  or  inverted,  according  to  the  place 
of  the  object. 

1.  The  object  between  the  mirror  and  its  principal  focus.  By 
Art.  588  (2),  rays  which  diverge  from  a  point  between  the  mirror 
and  its  principal  focus,  continue  to  diverge  after  reflection,  but  in 
a  less  degree.  Let  C  be  the  centre,  and  F  the  principal  focus  of 
the  mirror  M  N  (Fig.  312),  and  A  B  the  object.  Draw  the  axes, 


IMAGES    BY    THE    CONCAVE    MIRROR. 


357 


C  A,  C  B,  and  produce  them  behind  the  mirror.  The  pencil  from 
A  will  be  reflected  to  the  eye  at  H,  radiating  as  from  a,  in  the 
same  axis;  likewise,  those  from  B,  as  from  b.  Therefore,  the 


image  is  apparent,  since  rays  do  not  actually  flow  from  it ;  erect, 
as  the  axes  do  not  cross  each  other  between  the  object  and  image ; 
enlarged,  because  it  subtends  the  angle  of  the  axes  at  a  greater  dis- 
tance than  the  object  does.  As  the  object  approaches,  and  finally 
reaches  the  principal  focus,  the  reflected  rays  approach  parallelism, 
and  the  image  departs  from  the  mirror,  till  it  is  at  an  infinite  dis- 
tance, and  is  viewed  as  a  heavenly  body. 

2.  Object  between  the  principal  focus  and  the  centre.  As  soon 
as  the  object  passes  the  principal  focus,  the  rays  of  each  pencil  be- 
gin to  converge ;  and  each  radiant  of  the  object  has  its  conjugate 
focus  in  the  same  axis  beyond  the  centre  (Art.  589).  For  exam- 
ple, the  pencil  A  dg  (Fig.  313)  is  converged  to  a  in  the  axis  A  Ga, 
and  B  D  G  to  b,  in  the  axis  EC  I.  Therefore,  the  image  of  A  B 
is  a  b  beyond  the  centre ;  and  if  an  observer  is  beyond  a  b,  the 
rays,  after  crossing  at  the  image,  will  reach  him,  as  though  they 

FIG.  313. 


originated  in  a  b ;  or  if  a  screen  is  placed  at  a  b,  the  light  which 
is  collected  in  the  focal  points  will  be  thrown  in  all  directions  by 
radiant  reflection  from  the  screen.  Hence,  the  image  is  real ;  it 
is  also  inverted,  because  the  axes  cross  between  the  conjugate  foci ; 
and  it  is  enlarged,  since  it  subtends  the  angle  of  the  axes  at  a 
greater  distance  than  the  object  does.  That  I  O  is  greater  than 
B  C,  is  proved  by  joining  C  G,  which  bisects  the  angle  B  G  b,  and 
therefore  divides  B  b  so  that  B  C :  Cb  : :  B  G  :  G  b.  When  the 
object  reaches  the  centre,  the  image  is  there  also,  but  inverted  in 
position,  since  rays  which  proceed  from  one  side  of  C,  are  reflected 
to  the  other  side  of  it. 


358 


LIGHT. 


3.  Object  beyond  the  centre.  This  is  the  reverse  of  (2),  the 
conjugate  foci  having  changed  places ;  a  I),  therefore,  being  the 
object,  A  B  is  its  image,  real,  inverted,  diminished.  As  the  ob- 
ject removes  to  infinity,  the  image  proceeds  only  to  the  principal 
focus  F. 

600.  Illustrated  by  Experiment.— These  cases  are  shown 
experimentally  by  placing  a  lamp  close  to  the  mirror,  and  then 
carrying  it  along  the  axis  to  a  considerable  distance  away.    While 
the  lamp  moves  from  the  mirror  to  the  principal  focus,  its  image 
behind  the  mirror  recedes  from  its  surface  to  infinity ;  we  may 
then  regard  it  as  being  either  at  an  infinite  distance  behind,  or  an 
infinite  distance  in  front,  since  the  rays  of  every  pencil  are  par- 
allel.    After  the  lamp  passes  the  principal  focus,  the  image  ap- 
pears in  the  air  at  a  great  distance  in  front,  and  of  great  size,  and 
they  both  reach  the  centre  together,  where  they  pass  each  other  ; 
and,  as  the  lamp  is  carried  to  great  distances,  the  image,  growing 
less  and  less,  approaches  the  principal  focus,  and  is  there  reduced 
to  its  smallest  size.     The  only  part  of  the  infinite  line  of  the  axis 
before  and  behind,  in  which  no  image  can  appear,  is  the  small  dis- 
tance between  the  mirror  and  its  principal  focus. 

If  a  person  looks  at  himself,  so  long  as  he  is  between  the  mir- 
ror and  the  principal  focus,  he  sees  his  image  behind  the  mirror 
and  enlarged.  But  when  he  is  between  the  principal  focus  and 
centre,  the  image  is  real,  and  behind  him ;  the  converging  rays  of 
the  pencils,  however,  enter  his  eyes,  and  give  an  indistinct  view  of 
his  image  as  if  at  the  mirror.  When  he  reaches  the  centre,  the 
pupil  of  the  eye  is  seen  covering  the  entire  mirror,  because  rays 
from  the  centre  are  perpendicular,  and  return  to  it  from  all  parts 
of  the  surface.  Beyond  the  centre,  he  sees  the  real  image  in  the 
air  before  him,  distinct  and  inverted. 

601.  Images  by  the  Convex  Mirror. — The  convex  mirror 
affords  no  variety  of  cases,  because  diverging  rays,  which  fall  upon 

FIG.  314. 


it,  are  made  to  diverge  still  more  by  reflection.  -  In  Fig.  314,  the 
pencil  from  A  is  reflected,  as  if  radiating  from  a  in  the  same  axis 


SPHERICAL    ABERRATION    OF   MIRRORS.       350 

A  O}  and  that  from  B,  as  from  I  in  the  axis  B  C-  and  these  ap- 
parent radiants  are  always  nearer  the  surface  than  the  middle 
point  between  it  and  C  (Art.  590).  The  image  is  therefore  appar- 
ent; it  is  erect,  since  the  axes  do  not  cross  between  the  object  and 
image ;  and  it  is  diminished,  as  it  subtends  the  angle  of  the  axes 
at  a  less  distance  than  the  object 

602.  Caustics  by  Reflection. — These  are  luminous  curved 
surfaces,  formed  by  the  intersections  of  rays  reflected  from  a  hemi- 
spherical concave  mirror.      The.  name  caustic  is  given  from  the 
circumstance  that  heat,  as  well  as  light,  is  concentrated  in  the 
focal  points  which  compose  it.    BAD 

(Fig.  315),  represents  a  section  of  the  FlG-  315- 

mirror,  and  B  F  D  of  the  caustic  ;  the 
point  F,  where  all  the  sections  of  the 
caustic  through  the  axis  meet  each 
other,  is  called  the  cusp.  When  the 
incident  rays  are  parallel,  as  in  the 
figure,  the  cusp  is  at  the  principal 
focus,  that  is,  the  middle  point  be- 
tween A  and  C.  The  rays  near  the 
axis  R  A,  after  reflection  meet  at  the 
cusp  (Art.  588) ;  but  those  a  little  more 
distant  cross  them,  and  meet  the  axis  a  little  further  toward  A. 
And  the  more  distant  the  incident  ray  from  the  axis,  the  further 
from  the  centre  does  the  reflected  ray  meet  the  axis.  Thus  each 
ray  intersects  all  the  previous  ones,  and  this  series  of  intersections 
constitutes  the  curve,  B  F.  The  curve  is  luminous,  because  it 
consists  of  the  foci  of  the  successive  pencils  reflected  from  the 
arc  A  B. 

If  the  incident  rays,  instead  of  being  parallel,  diverge  from  a 
lamp  near  by,  the  form  of  the  caustic  is  a  little  altered,  and  the 
cusp  is  nearer  the  centre.  This  case  may  be  seen  on  the  surface 
of  milk,  the  light  of  the  lamp  being  reflected  by  the  edge  of  the 
bowl  which  contains  it. 

If  parallel  or  divergent  light  falls  on  a  convex  hemispherical 
mirror,  there  will  be  apparent  caustics  behind  the  mirror ;  that  is, 
the  light  will  be  reflected  as  if  it  radiated  from  points  arranged  in 
such  curves. 

603.  Spherical  Aberration  of  Mirrors.— It  has  already 
))een  mentioned  (Art.  587),  that  the  statements  in  this  chapter 
relating  to  focal  points  and  images,  as  produced  by  spherical  mir- 
rors, are  true  only  when  the  mirror  is  a  very  small  part  of  the 
whole  spherical  surface.    In  Art.  602  we  have  seen  the  effect  of 


360  LIGHT. 

using  a  large  part  of  the  spherical  surface — viz.,  the  rays  neither 
converge  to,  nor  diverge  from  a  single  point,  but  a  series  of  points 
arranged  in  a  curve.  This  general  effect  is  called  the  spherical 
aberration  of  a  mirror ;  since  the  deviation  of  the  rays  is  due  to 
the  spherical  curvature.  The  deviation,  as  we  have  seen,  is  quite 
apparent  in  a  hemisphere,  or  any  considerable  portion  of  one ;  but 
it  exists  in  some  degree  in  any  spherical  mirror,  unless  infinitely 
small  compared  with  the  hemisphere. 

But  there  are  curves  which  will  reflect  without  aberration. 
Let  a  concave  mirror  be  ground  to  the  form  of  a  paraboloid,  and 
rays  parallel  to  its  axis  will  be  converged  to  the  focus  without 
aberration.  For,  at  any  point  on  such  a  mirror,  a  line  parallel  to 
the  axis,  and  a  line  drawn  to  the  focus,  make  equal  angles  with 
the  tangent,  and  therefore,  equal  angles  with  the  perpendicular  to 
the  surface.  And  rays,  parallel  to  the  axis  of  a  convex  paraboloid, 
will  diverge  as  if  from  its  focus,  on  the  same  account.  Again,  if  a 
radiant  is  placed  at  the  focus  of  a  concave  parabolic  mirror,  the 
reflected  rays  will  be  parallel  to  the  axis,  and  will  illuminate  at  a 
great  distance  in  that  direction.  Such  a  mirror,  with  a  lamp  in 
its  focus,  is  placed  in  front  of  the  locomotive  engine  to  light  the 
track,  and  has  been  much  used  in  light-houses.  If  a  concave  mir- 
ror is  ellipsoidal,  light  emanating  from  one  focus  is  collected  with- 
out aberration  to  the  other,  because  lines  from  the  foci  to  any  point 
of  the  curve  make  equal  angles  with  the  tangent  at  that  point. 

Since  heat  is  reflected  according  to  the  same  law  as  light,  a 
concave  mirror  is  a  burning-glass.  When  it  faces  the  sun,  the 
light  and  heat  are  both  collected  in  a  small  image  of  the  sun  at 
the  principal  focus.  And,  if  no  heat  were  lost  by  the  reflection, 
the  intensity  at  the  focus  would  be  to  that  of  the  direct  rays,  as 
the  area  of  the  mirror  to  the  area  of  the  sun's  image.  Burning 
mirrors  have  sometimes  been  constructed  on  a  large  scale,  by  giv- 
ing a  concave  arrangement  to  a  great  number  of  plane  mirrors. 


CHAPTER   III. 

REFRACTION    OP    LIGHT. 

604.  Division  of  the  Incident  Beam.— When  light  falls 
on  an  opaque  body,  we  have  noticed  that  it  is  arrested,  and  a 
shadow  formed  beyond.  Of  the  light  thus  arrested,  a  portion  is 
reflected,  and  another  portion  lost,  which  is  said  to  be  absorbed  by 


KEFRACTION. 


361 


the  body.  When  light  meets  a  transparent  body,  a  part  is  still 
reflected,  and  a  small  portion  absorbed,  but,  in  general,  the  greater 
part  is  transmitted.  The  ratio  of  intensities  in  the  reflected  and 
transmitted  Beams  varies  with  the  angle  of  incidence,  but  little 
being  reflected  at  small  angles  of  incidence,  and  almost  the  whole 
at  angles  near  90°. 

605.  Refraction. — The  transmitted  beam  suffers  important 
changes,  one  of  which  is  a  change  in  direction.  This  change  is 
called  refraction,  and  takes  place  at  the  surface  of  a  new  medium. 
In  Fig.  316,  A  C,  incident  upon  R  S,  the  surface  of  a  different 
medium,  is  turned  at  C  into  another 
line,  as  C  E,  which  is  called  the  re-  FIG.  316. 

fracted  ray.  The  angle  E*C  Q,  be- 
tween the  refracted  ray  and  the  perpen- 
dicular is  called  the  angle  of  refraction  ; 
the  angle  O  C  E9  between  the  direc- 
tions of  the  incident  and  the  refracted 
rays,  is  the  angle  of  deviation. 

It  is  a  general  fact,  to  which  there 
are  but  few  exceptions,  that  a  ray  of 

light  in  passing  out  of  a  rarer  into  a  denser  medium  is  refracted 
toivard  the  perpendicular  to  the  surface ;  and  in  passing  out  of  a 
denser  into  a  rarer  medium,  it  is  refracted  from  the  perpendicular. 
But  the  chemical  constitution  of  bodies  sometimes  affects  their 
refracting  power.  Some  inflammable  bodies,  as  sulphur,  amber, 
and  certain  oils,  have  a  great  refracting  power  in  comparison  with 
other  bodies ;  and  in  a  given  instance,  a  ray  of  light  in  passing 
out  of  one  of  these  substances  into  another  of  greater  density  may 
be  turned  from  the  perpendicular  instead  of  toward  it.  In  the 
optical  use  of  the  words,  therefore,  denser  is  understood  to  mean, 
of  greater  refractive  power ;  and  rarer  signifies,  of  less  refractive 
power.  In  Fig.  316,  the  medium  below  R  8  is  of  greater  refrac- 
tive power  than  that  above. 

We  see  an  example  of  refraction  in  the  bent  appearance  of  an 
oar  in  the  water,  the  light  which  comes  to  the  eye  from  the  part 
immersed  is  bent  from  the  perpendicular  as  it  passes  from  water 
into  air,  and  causes  it  to  appear  higher  than  its  true  place.  In  the 
same  manner,  the  bottom  of  a  river  appears  elevated,  and  dimin- 
ishes the  apparent  depth  of  the  stream.  Let  a  small  object  be 
placed  in  the  bottom  of  a  bowl,  and  let  the  eye  be  withdrawn  till 
the  object  is  hidden  from  view  by  the  edge  of  the  bowl.  If  now 
the  bowl  be  filled  up  with  water,  the  object  is  no  longer  concealed, 
for  the  light,  as  it  emerges  from  the  water,  is  bent  away  from  the 
perpendicular,  and  brought  low  enough  to  enter  the  eye. 


362 


LIGHT. 


696.  Law  of  Refraction. — The  law  which  is  found  to  hold 
true  in  all  cases  of  common  refraction  is  this : 

The  angles  of  incidence  and  refraction  are  on  opposite  sides  of 
the  perpendicular  to  the  surface,  and,  for  any  given  ihedia,  the  sines 
of  the  angles  have  a  constant  ratio  for  all  inclinations. 

For  example,  in  Fig.  317,  if  A  C  is  refracted  to  E,  then  a  G 
will  be  refracted  to  e,  so  that  A  D :  E  F 
: :  a  d  :  ef\  and  if  the  rays  pass  out  in 
a  contrary  direction,  the  ratio  is  also 
constant,  being  the  reciprocal  of  the 
former,  viz.,  E  F:  A  D  ::  e  f:  a  d. 

A  ray  perpendicular  to  the  surface, 
passing  in  either  direction,  is  not  re- 
fracted; for,  according  to  the  law,  if 
the  sine  of  one  angle  is  zero,  the  sine  of 
the  other  must  be  zero  also.  Which- 
ever way  light  passes,  when  air  is  one 
of  the  media,  suppose  the  sine  of  the 
smaller  angle,  i.  e.  the  angle  in  the  denser  medium,  to  be  1,  then 
the  sine  of  the  larger  angle  for  water  is  1.336 ;  and  for  crown  glass, 
it  is  about  1.5.  The  number,  in  each  case,  expresses  the  constant 
ratio  of  the  sines,  for  the  given  media,  and  is  called  the  index  of 
refraction,  and  is  employed  as  the  measure  of  refractive  power. 
The  following  table  gives  the  index  of  refraction  for  a  few  sub- 
stances : 


Chromate  of  lead,   .    .    .  2.974 
Red  silver  ore,    .    •    .    .2.564 

Diamond, 2.439 

Phosphorus, 2.224 

Sulphur, 2.148 

Flint  glass, 1.830 

Sapphire, 1.800 

Sulphuret  of  carbon,     .    .1.768 

Oil  of  cassja, 1.641 

Quartz, 1.548 


Amber, *-547 

Crown  glass,   .....  1.530 

Oil  of  olives, 1.470 

Alum, 1-457 

Fluor  spar, 1-434 

Mineral  acids,      .    .    .    .1.410 

Alcohol, 1.372 

Water, 1.336 

Ice, I«3°9 

Tabasheer, i.m 


607.  Limit  of  Transmission  from 
a  Denser  to  a  Rarer  Medium.— As  a 

consequence  of  the  law  of  refraction, 
there  is  a  limit  beyond  which  a  ray  can- 
not escape  from  a  denser  medium.  Let 
A  C  (Fig.  318)  be  the  ray  incident  upon 
the  rarer  medium  RES.  It  will  be 
refracted  from  the  perpendicular  D  F 
into  the  direction  C  E,  so  that  A  D  is  to 


FIG.  318. 


LIGHT    THROUGH    PLANE    SURFACES. 


363 


E  F  in  a  constant  ratio  (Art.  606).  If  the  angle  A  C  D  be 
increased,  the  angle  F  C  E  must  also  increase  till  its  sine  equals 
C  S.  Make  a  d :  C  S :  :  A  D :  E  F\  then  a  C  is  the  limit  of  the  inci- 
dent rays  which  can  emerge.  For  if  a  CD  is  enlarged,  its  sine 
is  increased,  and  therefore  the  sine  of  refraction  must  increase; 
but  this  is  impossible,  since  it  is  already  equal  to  the  radius  C  S. 
Hence  it  follows,  that  whenever  the  angle  of  incidence  is  greater 
than  that  at  which  the  sine  of  the  angle  of  refraction  becomes 
equal  to  radius,  the  ray  cannot  be  refracted  consistently  with  the 
constant  ratio  of  the  sines. 

This  is  proved  also  by  experiment ;  the  emerging  ray  increases 
its  angle  of  refraction  till  it  at  length  ceases  to  pass. out.  Beyond 
that  limit  all  the  incident  rays  are  reflected  from  the  inner  surface 
of  the  denser  medium ;  and  this  reflection  is  more  perfect  than 
any  external  reflection,  and  is  called  total  reflection.  If  n  =  the 
index  of  refraction,  the  limit  at  whicn  refraction  ceases  and  total 
reflection  begins  is  found  by  the  proportion,  n  '.  1  : :  rad. :  sine  of 
the  limit.  If  the  refractive  power  is  greater,  the  limit  is  smaller ; 
for,  by  the  above  proportion,  since  the  means  are  constant,  n  varies 
inversely  as  sine  of  limit.  For  water,  it  is  48°  28' ;  for  crown  glass, 
40°  49';  for  diamond,  24°  12'. 

608.  Transmission  through  Plane  Surfaces. — 

1.  A  medium  bounded  by  parallel  planes.     In  this  case  the 
incident  and  emergent  rays  are  parallel.    Let  D  E  (Fig  319)  enter 
the  medium  A  B 1)  a  at  E,  and  leave 

it  at  F,  and  let  P  Q,  R  S  be  the 
perpendiculars  at  E,  F.-  The  first 
angle  of  refraction  Q  E  F,  and  the 
second  angle  of  incidence,  E  F  R, 
are  equal,  being  alternate;  there- 
fore, D  E  P  —  S  F  G,  since  their 
sines  have  a  constant  ratio  to  those 
of  Q  E  F,  E  F  R.  Hence,  if  the 
incident  rays  are  produced  to  G  and 

ff,  the  angles  of  deviation  are  equal ;  but  D  E  Fis  supplement  to 
the  first  angle  of  deviation,  and 
E  F  #  of  the  second.    There- 
alternate  and  equal,  D  E  is 
parallel  to  F  G. 

2.  A  medium  bounded  by 
inclined  planes,  called  a  prism. 
The  transmitted  ray  is  turned 
from  the  refracting  angle.   Let 


FlG  339. 


364  LIGHT. 

ABC  (Fig.  320)  be  that  section  of  a  glass  prism  which  is  per- 
pendicular to  its  axis,  and  A  C,  B  C,  the  inclined  sides  of  it, 
through  which  the  light  is  transmitted.  C  is  called  the  refracting 
angle,  and  A  B  the  base.  As  prisms  are  usually  constructed  and 
mounted,  either  A,  B,  or  C  may  be  the  refracting  angle ;  but  it  is 
not  essential  that  any  of  the  faces  should  meet  at  an  edge,  as  the 
effect  on  the  light  depends  only  on  the  inclination.  In  ordinary 
directions  of  the  ray,  the  two  refractions,  one  on  entering,  the 
other  on  leaving  the  prism,  conspire  to  increase  the  deviation  of 
the  ray  from  its  original  direction.  D  E  is  first  bent  toward  E  K, 
making  the  deviation  HE  F\  at  Fy\i  is  turned  fromF  Q,  making 
a  second  deviation,  E  F  /,  the  same  way.  The  sum  of  the  two 
deviations,  IEF+EFI=GIH,  the  total  deviation  away 
from  the  refracting  angle,  (7. 

To  an  eye  at  G,  the  radiant  D  is  seen  in  the  direction  G  F  I. 

609.  The   Multiplying   Glass. — A  piece  of  glass  ground 
with  one  side  plane,  and  the  other  in  any  number  of  plane  facets 
on  a  convex  surface,  is  called  a  multiplying  glass     Each  facet, 
along  with  the  opposite  plane  surface,  forms  a  prism;  and  if  a' 
radiant  A  is  placed  in  the  axis  (a  perpendicular  through  the  cen- 
tre of  the  plane  surface),  the 

pencils,  falling  on  the  several  FIG.  321. 

facets,  will  be  turned  from 

the    edge,  and  may  by  two 

refractions  at    the    opposite 

surfaces  be  brought  to  an  eye 

placed  also  in  the  axis,  and 

thus  as  many  images  will  be  ^^    .__ 

seen  as  there  are  facets.     Fig.  ""^-..  F"  p] 

321    exhibits    the    effect    of  tifi 

seven  such  facets. 

610.  Prism  used  for  Measuring  Refractive  Power.— 

The  following  theorem  may  be  used  for  determining  the  refractive 
power  of  a  substance,  after  first  forming  it  into  a  prism  of  small 
angle : 

If  the  angle  of  deviation  be  divided  ly  the  refracting  angle  of  the 
prism,  and  the  quotient  be  added  to  unity,  the  sum  is  the  index  of 
refraction. 

In  proving  this,  it  is  assumed  that  all  the  angles  are  very 
small,  so  that  they  vary  as  their  sines.  Let  n  —  the  index  of  re- 
fraction, then  (Fig.  320), 

X  E  I(=D  E  P)\K  E  F::  nil;  /.  FE  I:  KEF: :  n-l ;  1; 
also  KFI(=GFQ):  KFE\ :  n :  1 ;  /.  EFI\  KFE\ :  n-l :  I ; 


LIGHT    THROUGH    ONE    SURFACE.  355 

/.  F  E  I  +  E  F  I:  K  E  F  +  K  F  E\\n  —  \\\\ 
.-.  FIHiP'  XFi:n-l  •  1. 

But  P'  K  F  =  A  C  B,  each  being  the  supplement  of  E  K  F. 
Therefore,  Fill'.  A  CB  \\n-\\\\ 

_FIH  _FIH 

''          ~ 


Tjl    J     TT 

Now,  in  crown  glass,  is  found  by  trial  to  be  yery  near  J  ; 

A  C  Jj 

/.  n  —  1.5  nearly. 

In  order  to  find  the  index  of  refraction  for  any  solid  substance, 
grind  it  into  a  prism  whose  sides  are  nearly  parallel,  and  carefully 
measure  their  inclination.  Then  measure  the  displacement  of  a 
distant  object  seen  through  it  at  right  angles  to  its  surface.  For 
example,  the  faces  of  a  transparent  mineral  incline  1°  10';  and 
when  held  before  the  eye,  it  displaces  a  distant  object  50'  ;  /.  the 
index  of  refraction  =  1  +  f  =  1.714. 

611.  Light  through  one  Surface.  — 

1.  Plane  Surface.  When  parallel  rays  pass  into  another  me- 
dium through  a  plane  surface,  they  remain  parallel.  For  the  per- 
pendiculars being  parallel,  the  angles  of  incidence  are  equal,  and 
therefore  the  angles  of  refraction  are  equal  also,  and  the  refracted 
rays  parallel.  But  a  pencil  of  diverging  rays  is  made  to  diverge 
less,  when  it  enters  a  denser  medium.  For  the  outer  rays  make 
the  largest  angles  of  incidence,  and  are  therefore  most  refracted 
toward  the  perpendiculars,  and  thus  toward  parallelism  with  each 
other.  And  when  diverging  rays  enter  a  rarer  medium,  they  di- 
verge more  ;  because  the  outside  rays  make  the  largest  angles  of 
incidence,  and  therefore  the  largest  angles  of  refrac- 
tion,  by  which  means  they  spread  more  from  each 
other. 

The  last  case  is  illustrated  when  we  look  per- 
pendicularly into  water,  and  see  its  depth  apparently 
diminished  by  about  one-fourth  of  the  whole.  Let 
A  B  (Fig.  322)  be  the  surface,  and  C  a  point  at  the 
bottom,  from  which  a  pencil  comes  to  the  eye.  Let 
C  F,  the  axis  of  the  pencil,  be  perpendicular  to  A  B, 
and  C  B  E  an  oblique  ray  of  the  pencil.  The  an- 
gle C  —  C  B  H  =  angle  of  incidence;  and  A  D  B 
—  0  B  E  —  angle  of  refraction.  Now,  in  the  tri- 
angle B  D  C,  B  C-.  B  D  (:  .  A  C:  A  D  nearly)  .  : 
sin  D  :  sin  C  :  :  sine  of  refraction  ;  sine  of  incidence 
:  .  1.34  :  1.  Hence  the  apparent  depth  is  one-fourth 
less  than  the  real  depth.  The  apparent  depth  of 


366 


LIGHT. 


water  may  be  diminished  much  more  than  this  by  looking  into  it 
obliquely. 

2.  Convex  surface  of  the  denser.  A  convex  surface  tends  to 
converge  rays.  Let  C*  (Fig.  323)  be  the  centre  of  convexity,  and 
C'  D,  C'  0}  two  radii  produced.  As  rays  are  bent  toward  the  per. 


FIG.  323. 


pendiculars  in  entering  a  denser  medium,  and  as  the  perpendicu- 
lars themselves  converge  to  C',  the  general  effect  of  such  a  surface 
is  to  produce  convergency.  The  pencil,  A  H,  A  N,  is  merely 
made  less  divergent,  H  D',  N  A' ;  B  H,  B  N  become  parallel, 
H  D',  N  B' ;  D  H,  D  N,  convergent  to  D' ;  the  parallel  rays,  D  H, 
EN)  convergent  to  E{ ';  the  convergent  pencil,  D  H,  F N,  more 
convergent  to  F' \  but  D  H,  G  N,  which  converge  equally  with 
the  radii,  are  not  changed ;  and  D  H,  Gr  N,  which  converge  more 
than  the  radii,  converge  less  than  before,  to  G '.  The  two  last 
cases,  which  are  exceptions  to  the  general  effect,  rarely  occur  in 
the  practical  use  of  lenses. 

If  we  trace  in  the  opposite  direction  the  rays,  A ',  B',  Dr,  &c., 
comparing  each  with  D'  D,  we  find,  in  this  case  also,  that  the 
convex  surface  tends  to  converge  the  rays,  by  bending  them  from 
CTD,  C'  C. 

3.  Concave  surface  of  the  denser.  A  concave  surface  tends  to 
diverge  rays.  Let  C  C',  CD  (Fig.  324),  be  the  radii  of  concavity 
produced.  As  the  radii  diverge  in  the  direction  in  which  the  light 

FIG.  324. 


moves,  the  rays,  being  bent  toward  them,  will  generally  be  made 
to  diverge  also.  Hence,  parallel  rays,  B  H,  E  N,  are  diverged, 
H  Z),  N  E' ;  and  diverging  rays,  B  H,  B  N,  are  diverged  more, 
H  D,  N  B'.  If,  however,  rays  diverge  as  much  as  the  radii,  or 


LENSES. 


367 


more,  they  proceed  in  the  same  direction,  or  diverge  less,  a  case 
which  rarely  occurs. 

If  the  rays  are  traced  in  the  opposite  direction,  the  tendency 
in  general  to  produce  divergency  appears  from  the  fact  that  the 
perpendiculars  are  now  converging  lines,  and  the  rays  are  refracted 
from  them. 

612.  Lenses. — A  lens  is  a  circular  piece  of  glass,  whose  sur- 
faces are  plane  or  spherical,  and  the  spherical  surface  either  convex 
or  concave.  The  usual  varieties  are  shown  in  Pig.  325. 

FIG.  325. 


A  double  convex  lens  (A)  consists  of  two  spherical  segments, 
either  equally  or  unequally  convex,  having  a  common  hase. 

A  plano-convex  lens  (B)  is  a  lens  having  one  of  its  sides  con- 
vex and  the  other  plane,  being  simply  a  segment  of  a  sphere. 

A  double  concave  lens  (C)  is  a  solid  bounded  by  two  concave 
spherical  surfaces,  which  may  be  either  equally  or  unequally 
concave. 

A  plano-concave  lens  (D)  is  a  lens  one  of  whose  surfaces  is 
plane  and  the  other  concave. 

A  meniscus  (E)  is  a  lens  one  of  whose  surfaces  is  convex  and 
the  other  concave,  but  the  concavity  being  less  than  the  convexity, 
it  takes  the  form  of  a  crescent,  and  has  the  effect  of  a  convex  lens 
whose  convexity  is  equal  to  the  difference  between  the  sphericities 
of  the  two  sides. 

A  concavo-convex  lens  (F)  is  a  lens  one  of  whose  surfaces  is 
convex  and  the  other  concave,  the  concavity  exceeding  the  con- 
vexity, and  the  lens  being  therefore  equivalent  to  a  concave  lens 
whose  concavity  is  equal  to  the  difference  between  the  sphericities 
of  the  two  sides. 

A  line  (M  N}  passing  through  a  lens,  perpendicular  to  its  op- 
posite surfaces,  is  called  the  axis.  The  axis  usually,  though  not 
necessarily,  passes  through  the  centre  of  the  figure. 

613.  General  Effect  of  the  Convex  Lens. — Whether  dou- 
ble-convex or  plano-convex,  its  general  effect  is  to  converge  light. 
It  has  been  shown  (Art.  611)  that  the  convex  surface  of  a  denser 


368 


LIGHT. 


medium  tends  to  converge  rays,  whichever  way  they  pass  through 
it.    Therefore,  if  E  ("Fig.  326)  is  a  radiant,  while  E  C'  C  follows 


the  axis  without  change  of  direction,  the  oblique  ray  E  D  is  first 
refracted  toward  D  C,  and  then  from  C'  D'  produced,  and  both 
actions  conspire  to  converge  it  to  the  axis.  The  rays  are  repre- 
sented as  meeting  in  the  focus  F.  Whether  the  rays  are  actually 
converged,  depends  on  their  previous  relation  to  each  other.  If 
the  lens  is  plano-convex,  the  plane  surface  has  usually  but  little 
effect  in  converging  the  light ;  but  by  Art.  611  it  may  be  shown 
that  its  action  will  usually  conspire  with  that  of  the  convex 
surface. 

614.  General  Effect  of  the  Concave  Lens. — This  lens, 
whether  double-concave  or  plano-concave,  tends  to  produce  diver- 
gency. This  is  evident  from  what  has  been  shown  in  Art.  611. 
The  ray  ED  (Fig.  327),  in  entering  the  denser  medium,  is  first 

FIG.  327. 


refracted  totvard  C'  D  produced,  and  on  leaving  the  medium  at  7)', 
is  refracted  from  D  G ;  and  is  thus  twice  refracted  from  the  ray 
E  C,  which  being  in  the  axis,  is  not  refracted  at  all.  If  the  lens 
is  plano-concave,  the  effect  of  the  plane  surface  may,  or  may  not, 
conspire  with  that  of  the  concave  surface. 

615.  The  Optic  Centre  of  a  Lens.— Within  every  lens 
there  is  a  point  called  the  optic  centre,  so  situated  that  the  inci- 
dent and  emergent  portions  of  every  ray  which  passes  through  it 
are  parallel  to  each  other.  Let  C,  C'  (Fig.  328),  be  the  centres  of 
the  two  surfaces  of  the  lens ;  draw  the  axis  G  G',  also  any  oblique 


CONJUGATE    FOCI. 


369 


FIG.  328. 


C' 


radius  C  A,  and  C'B  parallel  to  it ;  then  join  A  B ;  the  point  E, 
in  which  A  B  intersects  the  axis,  is  the  optic  centre,  and  R  A  the 
incident,  and  B  11'  the  emer- 
gent portion  of  the  ray  passing 
through  A  and  B,  are  parallel 
to  each  other.  For  the  angles 
EAC,tmdiEB  C',  are  equal, 
being  alternate,  and  therefore 
the  ray  is  refracted  at  A  and  B 

equally  and  in  opposite  directions,  making  R  A  and  B  R  par- 
allel, as  proved  in  Art.  608,  1.  But  the  point  E  is  the  same, 
whatever  may  be  the  points  A  and  B,  to  which  the  parallel  radii, 
CA,  C1  B,  are  drawn.  For,  since  the  triangles,  EAC,EBC', 
are  similar,  GA  :  C'  B  : :  C  E :  C'  E;  .-.  CA  +  C'  B  :  C'  B  : :  CE 
+  C1  E  :  C'  E]  and  as  the  three  first  terms  are  constant,  the 
fourth,  (7  E,  is  constant  also,  and  E  is  a  fixed  point. 

When  the  lens  is  thin,  and  the  rays  are  nearly  parallel  to  its 
axis,  the  ray  R  A  B  R  may  be  considered  a  straight  line ;  and  it 
forms  the  axis  of  the  pencil  of  light  which  passes  through  the  lens 
in  that  direction. 


616.  Conjugate  Foci.— If  the  rays  from  R  (Fig.  329)  are 
collected  at  F,  then  rays  emanating  from  F  will  be  returned 
to  R]  and  the  two  points  are  called  conjugate  foci.  Their  rela- 
tive distances  from  the  lens  may  be  determined  when  the  radii  of 

FIG.  329. 


the  surfaces  and  the  index  of  refraction  are  known.  Let  n  be  the 
index  of  refraction,  and  assume,  what  is  practically  true,  that  the 
angles  of  incidence  and  refraction  are  so  small  that  their  ratio  is 
the  same  as  the  ratio  of  their  sines.  Then 


in  like  manner 

.-.KGH  + 
But        K 
and       I 
24 


KHG  :  IHG:  in  —  1:1; 
Q-.IQH+  IE  Gun 


=  C  +  C' 


370  LIGHT. 

naming  the  acute  angles   at  R,  C,  C',  F,  by  those  letters  re- 
spectively 

/.  R  +  F:C+  C'::n-I:I. 

Now,  the  lens  being  thin,  and  the  angles  7?,  C,  Cr,  and  F  very 
small,  the  same  perpendicular  to  the  axis,  at  L,  the  centre  of  the 
lens,  may  be  considered  as  subtending  all  those  angles.  Hence, 
each  angle  is  as  the  reciprocal  of  its  distance  from  L.  Let  R  L  — 
p'j  F  L  =  q;  C  L  =  r;  and  C'  L  =  r'.  Then  the  equation  above 
becomes 

MsM,::.-l:l; 

p      q    r      r' 

which  expresses  in  general  the  relation  of  the  conjugate  foci.    To 
.adapt  it  to  crown-glass,  call  n  =  f  ,  and  we  have 

l+U+l,::l:«. 
p      q    r      r 

617.  To  find  the  Principal  Focus.  —  The  radiant  from 
which  parallel  rays  come  is  at  an  infinite  distance.  Therefore, 
making  p  =  oc,  ,  and  the  distance  of  the  principal  focus  =  F,  we 

have  -  =  0,  and 
P 


r  r 

this 


(n  -l)(r  +  r')  ' 


V 

If  the  curvatures  are  equal,  F  =  —,  —  —  —  ;  for  crown-glass  this 

A  (n      1) 

becomes  F  —  r  ;  that  is,  the  principal  focus  of  a  double  convex 
lens  of  crown-glass,  having  equal  curvatures,  is  at  the  centre  of 
convexity. 

The  foregoing  formulae  are  readily  adapted  to  the  other  forms 
of  lens.  When  a  surface  is  plane,  its  radius  is  infinite,  and 

-,  or  —  =  0.    When  concave,  its  centre  is  thrown  upon  the  same 

side  as  the  surface,  and  its  radius  is  to  be  called  negative.  And  if 
the  focal  distance,  as  given  by  the  formula,  becomes  negative,  it  is 
understood  to  be  on  the  same  side  as  the  radiant;  that  is,  the 
focus  is  a  virtual  radiant. 

618.  Images  by  the  Convex  Lens.  —  The  convex  lens  forms 
a  variety  of  images,  whose  character  and  position  depend  on  the 
place  of  the  object.  If  it  is  at  the  principal  focus,  the  rays  of  every 


IMAGES  BY  THE  CONVEX  LENS. 


371 


pencil  pass  out  parallel,  and  seem  to  come  from  an  infinite  dis- 
tance. If  the  object  is  nearer  than  the  principal  focus,  the  emer- 
gent rays  of  each  pencil  diverge  less  than  the  incident  rays,  and 
therefore  seem  to  radiate  from  points  further  back ;  the  image  is 
therefore  apparent.  Let  M  N  (Fig.  330)  be  the  object  nearer  than 
the  principal  focus,  F.  Then  the  pencil  from  If  will,  after  refrac- 

FIG.  330.  < 


tion,  diverge  as  from  mm  0  M  produced,  and  so  of  every  point ; 
hence  m  n  is  the  image.  It  is  erect,  because  the  axes  of  the  pencils 
do  not  cross  between  the  object  and  image ;  and  it  is  enlarged, 
because  it  subtends  the  angle  M  0  N  at  a  greater  distance  than 
the  object. 

But  if  the  object  is  further  from  the  lens  than  the  principal 
focus,  the  rays  of  each  pencil  converge  to  a  point  in  the  axis  of 
that  pencil  produced  through  the  lens ;  and  thus  light  is  collected 
in  focal  points,  which  consequently  become  actual  radiants.  The 
last  case  is  illustrated  by  Fig.  331,  in  which  M  Nis  the  object,  and 

Fia  331. 


m  n  the  image.  A  cone  of  rays  from  0  covers  the  lens  L  L,  and 
is  converged  again  into  the  axis  at  o,  the  conjugate  focus  of  0,  and 
there  cross,  and  proceed  as  from  a  radiant.  The  cone  of  rays  from 
M  is  converged  to  m  in  the  axis  Mm  of  that  cone,  which  is  a 
straight  line  through  the  optic  centre  (Art.  615) ;  and  so  from 
every  point  of  the  object.  Though  the  rays  of  every  radiant  con- 
verge from  the  lens  to  the  conjugate  focus  of  that  radiant,  yet  the 
axes  of  the  pencils  diverge  from  each  other,  having  all  crossed  at 
the  optic  centre.  The  image  is  therefore  inverted,  as  are  all  real 
images,  in  whatever  way  produced. 


LIGHT 


The  formula  for  conjugate  foci  shows  that  if  p  is  increased,  q 
is  diminished ;  therefore  the  further  M  N  is  removed  from  the 
lens,  the  nearer  m  n  approaches  to  it ;  but  the  nearest  position  is 
the  principal  focus,  which  it  reaches  when  the  object  is  at  an  infi- 
nite distance.  As  the  object  and  image  subtend  equal  angles  at 
the  optic  centre,  and  are  parallel,  or  nearly  parallel  with  each 
other,  their  diameters  are  proportioned  to  their  distances  from  the 
'lens.  But  the  area  of  the  lens  has  no  effect  on  the  size  of  the 
image,  since  change  of  area  does  not  alter  the  relation  of  the  axes, 
but  only  the  size  of  the  luminous  cones,  and  thus  the  quantity  of 
light  in  each  pencil. 

619.  Images  by  the  Concave  Lens. — As  the  rays  of  each 
pencil  are  diverged  more  after  passing  through  the  lens  than  before, 
the  image  is  apparent,  and  is  situated  between  the  lens  and  the 
object.  Let  M  N  (Fig.  332)  be  the  object;  the  cone  of  rays  from 
JVwill,  after  refraction,  diverge  more,  as  from  n,  in  the  same  axis 

FIG.  332. 


0  N;  and  all  other  pencils  will  be  affected  in  a  similar  manner, 
and  form  an  apparent  image  m  n.  It  will  be  erect,  since  the  axes 
do  not  cross  between,  and  diminished,  being  nearer  the  angle  (7, 
which  is  subtended  by  both  object  and  image. 

It  is  noticeable  that  the  concave  mirror  and  the  convex  lens  are 
analogous  in  their  effects,  forming  images  on  both  sides,  both  real 
and  apparent,  both  erect  and  inverted,  both  larger  and  smaller 
than  the  object ;  while  the  convex  mirror  and  the  concave  lens  also 
resemble  each  other,  producing  images  always  on  one  side,  always 
apparent,  always  erect,  always  smaller  than  the  object. 

620.  Caustics  by  Refraction. — If  the  convex  surface  of  a 
lens  is  a  considerable  part  of  a  hemisphere, 
the  rays  more  distant  from  the  axis  will  be 
so  much  more  refracted  than  others,  as  to 
cross  them  and  meet  the  axis  at  nearer 
points,  thus  forming  caustics  by  refraction. 
Fig.  333  shows  this  effect  in  the  case  of 
parallel  rays;  those  near  the  axis  inter- 


FIG.  333. 


REMEDY    FOR    SPHERICAL    ABERRATION.      373 

secting  it  at  the  principal  focus  F,  and  the  intersections  of  re- 
moter rays  being  nearer  and  nearer  to  the  lens,  so  that  the  whole 
converging  pencil  assumes  a  form  resembling  a  cone  with  concave 
sides. 

621.  Spherical  Aberration  of  a  Lens. — The  production 
of  caustics  is  an  extreme  case  of  what  is  called  spherical  aberra- 
tion.   Unless  the  lens  is  of  small  angular  breadth,  a  pencil  whose 
rays  originated  in  one  point  of  an  object  is  not  converged  accu- 
rately to  one  point  of  the  image,  but  the  outer  rays  are  refracted 
too  much,  and  make  their  focus  nearer  the  lens  than  that  of  the 
central  rays,  as  represented  in  Fig.  334.    If  F  is  the  focus  of  the 
central  rays,  and  F'  of  the 

extreme  ones,  other  rays  of  FlG>  334> 

the  same  beam  are  collected 
in  intermediate  points,  and 
F  F'  is  called  the  longitu- 
dinal spherical  aberration; 
and  G  ff,  the  breadth  cov- 
ered  by  the  pencil  at  the 
focus  of  central  rays,  is  called  the  lateral  spherical  aberration. 

Such  a  lens  cannot  form  a  distinct  image  of  any  object ;  be- 
cause perfect  distinctness  requires  that  all  rays  from  any  one  point 
of  the  object  should  be  collected  to  one  point  in  the  image.  If, 
for  example,  the  beam  whose  outside  rays  are  R  A,  R  B,  comes 
from  a  point  of  the  moon's  disk,  that  point  will  not  be  perfectly 
represented  by  F,  because  a  part  of  its  light  covers  the  circle, 
whose  diameter  is  G  H,  thus  overlapping  the  space  representing 
adjacent  points  of  the  moon.  And  if  that  point  had  been  on  the 
edge  of  the  moon's  disk,  F  could  not  be  a  point  of  a  well-defined 
edge  of  the  image,  since  a  part  of  the  light  would  be  spread  over 
the  distance  F  G  outside  of  it,  and  destroy  the  distinctness  of  its 
outline. 

622.  Remedy  for  Spherical  Aberration. — -As  spherical 
lenses  refract  too  much  those  rays  which  pass  through  the  outer 
parts,  it  is  obvious  that,  to  destroy  aberration,  a  lens  is  required 
whose  curvature  diminishes  toward  the  edges.   Accordingly,  forms 
for  ellipsoidal  lenses  have  been  calculated,  which  in  theory  will 
completely  remove  this  species  of  aberration.     But  no  curved 
solids  can  be  so  accurately  ground  as  those  whose  curvature  is  uni- 
form in  all  planes,  that  is,  the  spherical.    Hence,  in  practice  it  is 
found  better  to  reduce  the  aberration  as  much  as  possible  by  spher- 
ical lenses,  than  to  attempt  an  entire  removal  of  it  by  other  forms 
which  cannot  be  well  made. 


374  LIGHT. 

In  a  plano-convex  lens,  whose  plane  surface  is  toward  the  ob- 
ject, the  spherical  aberration  is  4.5;  that  is  (Fig.  334),  F  F'  —  4.5 
times  the  thickness  of  the  lens.  But  the  same  lens,  with  its  con- 
vex side  toward  the  object,  is  far  better,  its  aberration  being  only 
1.17.  In  a  double  convex  lens  of  equal  curvatures,  the  aberration 
is  1.67;  if  the  radii  of  curvature  are  as  1  :  6,  and  the  most  convex 
side  is  toward  the  object,  the  aberration  is  only  1.07.  By  placing 
two  plano-convex  lenses  near  each  other,  the  aberration  may  be 
still  more  reduced. 

623.  Atmospheric   Refraction. — The  atmosphere  may  be 
regarded  as  a  transparent  spherical  shell,  whose  density  increases 
from  its  upper  surface  to  the  earth.     The  radii  of  the  earth  pro- 
duced are  the  perpendiculars  of  all  the  laminse  of  the  air ;  and 
rays  of  light  coming  from  the  vacuum  beyond,  if  oblique,  are  bent 
gradually  toward  these  perpendiculars;    and  therefore  heavenly 
bodies  appear  more  elevated  than  they  really  are.    The  greatest 
elevation  by  refraction  takes  place  at  the  horizon,  where  it  is 
about  half  a  degree. 

624.  Mirage. — This  phenomenon,  called  also  looming,  con- 
sists of  the  formation  of  one  or  more  images  of  a  distant  object, 
caused  by  horizontal  strata  of  air  of  very  different  densities.    Ships 
at  sea  are  sometimes  seen  when  beyond  the  horizon,  and  their 
images  occasionally  assume  distorted  forms,  contracted  or  elon- 
gated in  a  vertical  direction.    These  effects  are  generally  ascribed 
to  extraordinary  refraction  in  horizontal  strata,  whose  difference 
of  density  is  unusually  great.    But  many  cases  of  mirage  seem  to 
be  instances  of  total  reflection  from  a  highly  rarefied  stratum  rest- 
ing on  the  earth.     These  occur  frequently  on  extended  sandy 
plains,  as  those  of  Egypt.    When  the  surface  becomes  heated, 
distant  villages,  on  more  elevated  ground,  are  seen  accompanied 
by  their  images  inverted  below  them,  as  in  water.    As  the  traveler 
advances,  what  appeared  to  be  an  expanse  of  water  retires  before 
him.    By  placing  alcohol  upon  water  in  a  glass  vessel,  and  allow- 
ing them  time  to  mingle  a  little  at  their  common  surface,  the  phe- 
nomena of  mirage  may  be  artificially  represented. 


THE     PRISMATIC    SPECTRUM. 


375 


CHAPTER   IV. 


DECOMPOSITION  AND  DISPERSION  OF  LIGHT. 

625.  The  Prismatic  Spectrum. — Another  change  which 
light  suffers  in  passing  into  a  new  medium,  is  called  decomposi- 
tion, or  the  separation  of  light  into  colors.  For  this  purpose,  the 
glass  prism  is  generally  employed.  It  is  so  mounted  on  a  jointed 
stand,  that  it  can  be  placed  in  any  desired  position  across  the 
beam  from  the  heliostat.  The  beam,  as  already  noticed,  is  bent 
away  from  the  refracting  angle,  both  in  entering  and  leaving  the 
prism,  and  deviates  several  degrees  from  its  former  direction.  If 
the  light  is  admitted  through  a  narrow  aperture,  F  (Fig.  335),  and 

FIG.  335. 


the  axis  of  the  prism  is  placed  parallel  to  the  length  of  the  aper- 
ture, the  light  no  longer  falls,  as  before,  in  a  narrow  line,  L,  but 
is  extended  into  a  band  of  colors,  R  V,  whose  length  is  in  a  plane 
at  right  angles  to  the  axis  of  the  prism.  This  is  called  the  pris- 
matic spectrum.  Its  colors  are  usually  regarded  as  seven  in 
number — -red,  orange,  yellow,  green,  Hue,  indigo,  violet.  The  red 
is  invariably  nearest  to  the  original  direction  of  the  beam,  and 
the  violet  the  most  remote;  and  it  is  because  the  elements  of 
white  light  are  unequally  refrangible,  that  they  become  separated, 
by  transmission  through  a  refracting  body.  The  spectrum  is 
properly  regarded  as  consisting  of  innumerable  shades  of  color. 
Instead  of  Newton's  division  into  seven  colors,  many  choose  to 
consider  all  the  varieties  of  tint  as  caused  by  the  combination  of 
three  primitive  colors,  red,  yellow,  and  blue,  varying  in  their  pro- 


376 


LIGHT. 


portions  throughout  the  entire  spectrum.  The  number  seven, 
as  perhaps  any  other  particular  number,  must  be  regarded  as 
arbitrary. 

The  spectrum  contains  other  elements  besides  those  which 
affect  the  eye.  If  a  thermometer  bulb  be  moved  along  the  spec- 
trum, it  is  found  that  the  greatest  heat  lies  outside  of  the  visible 
spectrum  at  the  red  extremity ;  hence  heat  is  less  refrangible  than 
light.  On  the  other  hand,  the  chemical  or  actinic  rays  are  more 
refrangible  than  the  luminous  rays,  and  fall  at  and  beyond  the 
violet  end  of  the  spectrum. 

Light  from  other  sources  is  also  susceptible  of  decomposition 
by  the  prism ;  but  the  spectrum,  though  resembling  that  of  the 
sun,  usually  differs  in  the  proportion  of  the  colors. 

626.  The  Individual  Colors  of  the  Spectrum  cannot  be 
Decomposed  by  Refraction.— Some  of  the  colors  of  the  spec- 
trum are  called  simple  colors,  namely,  red,  yellow,  and  blue,  while 
the  others  are  generally  regarded  as  compound  colors ;  for  orange 
may  be  formed  of  red  and  yellow,  green  of  yellow  and  blue,  while 
indigo  and  violet  are  mixtures  of  blue  and  red  in  different  propor- 
tions. It  is  nevertheless  true,  that  none  of  the  colors  of  the  spec- 
trum can  be  decomposed  by  refraction.  For  if  the  spectrum 
formed  by  the  prism  A  be  allowed  to  fall  on  the  screen  ED  (Fig. 
336),  and  one  color  of  it,  green  for  example,,  be  let  through  the 

FIG.  336. 


screen,  and  received  on  a  second  prism,  B,  it  is  still  refracted  as 
before,  but  all  its  rays  remain  together  and  of  the  same  color. 
The  same  is  true  of  every  color  of  the  spectrum.  Therefore,  so 
far  as  refrangibility  is  concerned,  all  the  colors  of  the  spectrum 
are  alike  simple. 

627.  Colors  of  the  Spectrum  Recombined. — It  may  be 

shown,  in  several  ways,  that  if  all  the  colors  of  the  spectrum  be 
combined,  they  will  reproduce  ivMte  light.  One  method  is  by 
transmitting  the  beam  successively  through  two  prisms  whose 
refracting  angles  are  on  opposite  sides.  By  the  first  prism,  the 
colors  are  separated  at  a  certain  angle  of  deviation,  and  then  fall 


FRAUNHOFER    LINES. 


377 


on  the  second,  which  tends  to  produce  the  same 
deviation  in  the  opposite  direction,  by  which 
means  all  the  colors  are  brought  upon  the  same 
ground,  and  the  illuminated  spot  is  white  as  if  no 
prism  had  been  interposed.  Or  the  colors  may 
be  received  on  a  series  of  small  plane  mirrors, 
which  admit  of  such  adjustment  as  to  reflect  all 
the  beams  upon  one  spot.  Or  finally,  the  several 
colors  can,  by  different  methods,  be  passed  so  rap- 
idly before  the  eye  that  their  visual  impressions 
shall  be  united  in  one ;  in  which  case  the'  illu- 
minated surface  appears  white. 

628.  Complementary  Colors. — If  certain 
colors  of  the  spectrum  are  combined  in  a  com- 
pound color,  and  the  others  in  another,  these  two 
are  called  complementary  colors,  because,  when 
united,  they  will  produce  white.    For  example,  if 
green,  Hue,  and  yellow  are  combined,  they  will 
produce  green,  differing  slightly  from  that  of  the 
spectrum ;  the  remaining  colors,  red,  orange,  in- 
digo, and  violet,  compose  a  kind  of  purple,  unlike 
any  color  of  the  spectrum.    But  these  particular 
shades  of  green  and  purple,  if  mingled,  will  make 
perfectly  white  light,  and  are  therefore  comple- 
mentary colors. 

629.  Fixed  Dark  Lines  of  the  Spectrum. 

— Let  the  aperture  through  which  the  sunbeam 
enters  be  made  exceedingly  narrow,  and  let  the 
prism  be  of  uniform  density,  and  then  let  the  re- 
fracted pencil  pass  immediately  through  a  small 
telescope,  and  thence  into  the  eye,  and  there  ap- 
pears a  phenomenon  of  great  interest — the  dark 
lines,  or  the  Fraunhofer  lines,  as  they  are  often 
called  from  the  name  of  their  discoverer.  These 
lines,  an  imperfect  view  of  which  is  presented  in 
Tig.  337,  are  unequal  in  breadth,  in  darkness,  and 
in  distance  from  each  other,  and  so  fine  and 
crowded  in  many  parts  that  the  whole  number 
cannot  be  counted.  Fraunhofer  himself  described 
between  500  and  600,  among  which  a  few  of  the 
most  prominent  are  marked  by  letters,  and  used 
in  measuring  refractive  power.  At  least  as  many 
as  six  thousand  are  now  known  and  mapped,  so 


FIG.  337. 

RED. 


378  LIGHT. 

that  any  one  of  them  may  be  identified.  They  are  parallel  to  each 
other,  and  perpendicular  to  the  length  of  the  spectrum.  When 
the  pencil  passes  through  a  succession  of  prisms,  all  bending  it 
the  same  way,  the  spectrum  becomes  more  dilated,  and  more  lines 
are  seen.  The  instrument  fitted  up  as  above  described,  either  with 
one  prism  or  a  series  of  prisms,  is  called  a  spectroscope. 

630.   Bright  Lines  in  the  Spectrum  of  Flame. — If  the 

spectroscope  be  used  for  the  examination  of  the  flame  of  different 
substances  in  combustion,  the  spectrum  is  found  to  consist  of  cer- 
tain bright  lines,  differing  in  color  and  number,  according  to  the 
substance  under  examination.  Thus,  the  spectrum  of  sodium 
flame,  besides  showing  other  fainter  lines,  consists  mainly  of  two 
conspicuous  yellow  lines,  very  close  together,  so  as  ordinarily  to 
appear  as  one.  The  flame  of  carbon  shows  two  distinct  lines,  one 
of  which  is  green,  the  other  indigo.  In  this  respect  every  sub- 
stance differs  from  every  other,  and  each  may  be  as  readily  distin- 
guished by  the  lines  which  compose  its  spectrum  as  by  any  other 
property.  The  lines  of  some  substances  are  very  numerous ;  as, 
for  example,  iron,  whose  spectrum  lines  amount  to  four  or  five 
hundred. 

But  a  solid  or  liquid  substance,  when  raised  to  a  red  or  white 
heat,  without  passing  into  the  gaseous  state  and  producing  flame, 
forms  a  continuous  spectrum,  having  neither  bright  nor  dark 
lines. 

631.  The  Spectrum  of  a  Heated  Solid  or  Liquid  Shin- 
ing through  Flame. — The  condition  of  a  spectrum  is  entirely 
changed  when  the  light  from  a  heated  solid  or  liquid  substance 
shines  through  the  flame  of  a  burning  gas.     The  bright  lines 
instantly  become  dark  lines.     The  flame  seems  to  absorb  just 
those  rays,  and  only  those,  which  are  like  the  rays  emitted  by 
itself.     As  an  example,  the  spectrum  of  sodium  flame  consists  of  a 
bright  double  yellow  line,  and  a  few  fine  luminous  lines  of  other 
colors.    Jf  now  iron  at  an  intense  white  heat  shines  through  this 
flame,  the  whole   spectrum  becomes  luminous,  except  the  very 
lines  which  were  before  bright ;  these  are  now  dark. 

632.  Composition  of  the  Sun's  Surface.— A  great  number 
of  the  dark  lines  of  the  solar  spectrum  are  identical  in  position 
with  lines  in  the  spectrum  of  terrestrial  substances.     The  spectro- 
scope can  be  attached  to  the  eye-piece  of  a  telescope,  so  as  to  bring 
half  the  breadth  of  the  solar  spectrum  side  by  side  with  half  the 
breadth  of  the  spectrum  of  the  flame  of  some  substance ;  and  their 
lines  can  thus  be  compared  with  each  other  on  the  divisions  of  the 
same  scale.    When  this  is  done,  there  is  found,  with  regard  to 


CHROMATIC    ABERRATION    OF    LENSES.         379 

several  substances,  an  identity  of  position  and  relative  breadth  and 
intensity  so  exact  that  it  is  impossible  to  regard  the  agreement  as 
accidental.  The  double  line  D  of  the  sunbeam,  is  the  prominent 
line  of  sodium.  So  all  the  numerous  lines  of  potassium,  iron,  and 
several  other  simple  substances,  exactly  coincide  with  the  dark 
lines  of  the  spectrum  of  sunlight. 

The  foregoing  facts  seem  to  indicate  that  the  photosphere  of 
the  sun  consists  of  the  flame  of  many  substances,  among  which 
are  some  such  as  belong  to  the  earth,  namely,  sodium,  potassium, 
iron,  &c. ;  and  that  the  luminous  liquid  matter  beneath  the  pho- 
tosphere shines  through  it,  and  changes  all  the  bright  lines  to 
dark  ones. 

633.  Dispersion  of  Light. — Decomposition  of  light  refers  to 
the  fact  of  a  separation  of  colors ;  dispersion,  rather  to  the  meas- 
ure or  degree  of  that  separation.  The  dispersive  power  of  a  me- 
dium indicates  the  amount  of  separation  which  it  produces,  com- 
pared with  the  amount  of  refraction.  For  example,  if  a  substance, 
in  refracting  a  beam  of  light  1°  51'  from  its  course,  separates  the 
violet  from  the  red  by  4',  then  its  dispersive  power  is  Tf  T  =  .036. 
The  following  table  gives  the  dispersive  power  of  a  few  substances 
much  used  in  optics : 


.Dispersive  power. 

Oil  of  Cassia,  .  .  0.139 
Sulphuret  of  Carbon,  0.130 
Oil  of  Bitter  Almonds,  0.079 
Flint-Glass,  .  .  .  0.052 
Muriatic  Acid,  .  .  0.043 
Diamond,  ....  0.038 


Dispersive  power. 

Plate-Glass,    .     .     .  0.032 

Sulphuric  Acid,  .     .  0.031 

Alcohol,     ....  0.029 

Rock-Crystal,     .     .  0.026 

Blue  Sapphire,    .     .  0.026 

Fluor-spar,     .     .     .  0.022 


Crown-Glass,  .     .     .     0.036 

The  discovery  that  different  substances  produce  different  de- 
grees of  dispersion,  is  due  to  Dollond,  who  soon  applied  it  to  the 
removal  of  a  serious  difficulty  in  the  construction  of  optical  in- 
struments. 

634.  Chromatic  Aberration  of  Lenses. — This  is  a  devia- 
tion of  light  from  a  focal  point,  occasioned  by  the  different  re- 
frangibility  of  the  colors.  If  the  surface  of  a  lens  be  covered, 
except  a  narrow  ring  near  the  edge,  and  a  sunbeam  be  transmitted 
through  the  ring,  the  chromatic  aberration  becomes  very  apparent ; 
for  the  most  refrangible  color,  violet,  comes  to  its  focus  nearest, 
and  then  the  other  colors  in  order,  the  focus  of  red  being  most 
remote.  Since  the  distinctness  of  an  image  depends  on  the  ac- 
curate meeting  of  rays  of  the  same  pencil  in  one  point,  it  is  clear 
that  discoloration  and  indistinctness  are  caused  by  the  separation 
of  colors. 


380 


LIGHT. 


635.  Achromatism. — In  order  to  refract  light,  and  still  keep 
the  colors  united,  it  is  necessary  that,  after  the  beam  has  been 
refracted,  and  thus  separated,  a  substance  of  greater  dispersive 
power  should  be  used,  which  may  bring  the  colors  together  again, 
by  refracting  the  beam  only  a  part  of  the  distance  back  to  its 
original  direction.  For  instance,  suppose  two  prisms,  one  of 
crown-glass  and  one  of  flint-glass,  each  ground  to  such  a  refract- 
ing angle  as  to  separate  the  violet  from  the  red  ray  by  4'.  In 
order  for  this,  the  crown-glass,  whose  dispersive  power  is  .036, 

4' 

must  refract  the  beam  1°  51'  j  for  ^0      ,  =  .036 ;  and  the  flint- 
glass,  whose  dispersive  power  is  .052,  must  refract  only  1°  17' ; 

4' 
f°r  -10 17/  =  -052.    Place  these  two  prisms  together,  base  to  edge, 

as  in  Fig.  338,  C  being  the  crown-glass  and  F  the  flint-glass.   Then 

FIG.  338. 


£7  will  refract  the  beam,  bb,  downward  1°  51',  and  the  violet,  v,  4r 
more  than  the  red,  r  ;  Fwill  refract  this  decomposed  beam  up- 
ward 1°  17',  and  the  violet  4'  more  than  the  red,  which  will  just 
bring  them  together  at  v  r.  Thus  the  colors  are  united  again,  and 
yet  the  beam  is  refracted  downward  1°  51'  —  1°  17'  =  34',  from  its 
original  direction. 

636.  Achromatic  Lens.— If  two  prisms  can  thus  produce 
achromatism,  the  same  may  be  effected  by  lenses ;  for  a  convex 
lens  of  crown-glass  may  converge  the  rays  of  a  pencil,  and  then  a 


FIG 


c 


concave  lens  of  flint-glass  may  diminish  that  convergency  suffi- 
ciently to  unite  the  colors.    A  lens  thus  constructed  of  two  lenses 


THE    RAINBOW.  381 

of  different  materials  and  opposite  curvatures,  so  adapted  as  to 
produce  an  image  free  from  chromatic  aberration,  is  called  an 
achromatic  lens.  Fig.  339  shows  such  a  combination.  The  convex 
lens  of  crown-glass  alone  would  gather  the  rays  into  a  series  of 
colored  foci  from  v  to  r ;  the  concave  flint-glass  lens  refracts  them 
partly  back  again,  and  collects  all  the  colors  at  one  point,  F. 

637.  Colors  not  Dispersed  Proportionally. — It  is  assumed 
in  the  foregoing  discussion,  that  when  the  red  and  violet  are 
united,  all  the  intermediate  colors  will  be  united  also.  It  is  found 
that  this  is  not  strictly  true,  but  that  different  substances  separate 
two  given  colors  of  the  spectrum  by  intervals  which  have  different 
ratios  to  the  whole  length  of  the  spectrum.  This  departure  from 
a  constant  ratio  in  the  distances  of  the  several  colors,  as  dispersed 
by  different  media,  is  called  the  irrationality  of  dispersion.  In 
consequence  of  it  there  will  exist  some  slight  discoloration  in  the 
image,  after  uniting  the  extreme  colors.  It  is  found  better  in 
practice  to  fit  the  curvatures  of  the  lenses,  for  uniting  those  rays 
which  most  powerfully  affect  the  eye. 


CHAPTER    V. 

RAINBOW    AND    HALO. 

638.  The   Rainbow. — This  phenomenon,  when  exhibited 
most  perfectly,  consists  of  two  colored  circular  arches,  projected  on 
falling  rain,  on  which  the  sun  is  shining  from  the  opposite  part 
of  the  heavens.    They  are  called  the  inner  or  primary  bow,  and 
the  outer  or  secondary  bow.    Each  contains  all  the  colors  of  the 
spectrum,  arranged  in  contrary  order ;  in  the  primary,  red  is  out- 
ermost ;  in  the  secondary,  violet  is  outermost.     The  primary  bow 
is  narrower  and  brighter  than  the  secondary,  and,  when  of  unusual 
brightness,  is  accompanied  by  supernumerary  bows,  as  they  are 
called ;  that  is,  narrow  red  arches  just  within  it,  or  overlapping 
the  violet ;  sometimes  three  or  four  supernumeraries  can  be  traced 
for  a  short  distance.    The  common  centre  of  the  bows  is  in  a  line 
drawn  from  the  sun  through  the  eye  of  the  spectator. 

639.  Action  of  a  Transparent  Sphere  on  Light— It  will 
aid  in  understanding  the  manner  in  which  the  bow  is  formed,  to 
notice  the  experiments  which  first  led  to  a  correct  theory  respect- 
ing it.    Let  a  hollow  sphere  of  glass  be  filled  with  water  and  placed 


382  LIGHT. 

in  the  sunlight,  and  then  let  the  directions  and  conditions  of  the 
most  luminous  pencils  which  emerge  from  it  be  observed.  Fig. 
340  exhibits  the  general  result.  The  entire  hemisphere  exposed 
to  the  rays  is  of  course  penetrated  by  them ;  but  a  narrow  pencil, 
S  Ay  about  60°  distant  from  S  C,  the  axis 
of  the  drop,  that  is,  the  ray  which  passes 
through  its  centre,  is  converged  to  B, 
where  some  light  escapes,  but  a  large  part 
is  reflected  to  D ;  at  that  point  a  division 
occurs  again,  and  the  emerging  pencil, 
D  E,  consists  of  decomposed  light,  each 
color  of  which  can  be  seen  at  a  great  dis- 
tance. The  part  reflected  at  D  divides 
again  near  F,  but  the  emergent  portion  diverges,  and  has  not  the 
intensity  of  D  K  But  if  a  pencil,  which  strikes  the  drop  at  a 
about  10°  outside  of  8  A,  be  traced,  we  find  that  a  part  is  reflected 
near  B,  a  part  of  that  again  reflected  at  d,  and  then,  on  reaching 
the  point  F9  the  emerging  portion  is  not  only  decomposed,  but  re- 
tains its  intensity  to  a  great  distance.  The  pencil  D  E,  which  has 
been  once  reflected  at  B,  is  the  one  concerned  in  the  production 
of  the  primary  bow.  The  pencil  F  Gr,  which  leaves  the  drop  at 
F,  after  it  has  been  twice  reflected,  is  one  of  those  which  consti- 
tute the  secondary  bow. 

Most  of  the  light  which  enters  a  drop  of  rain,  and  leaves  it 
again,  either  not  reflected  at  all,  or  reflected  one  or  more  times,  is 
scattered  in  various  directions,  and  brings  to  the  eye  of  an  ob- 
server no  impression  of  intense  light.  It  is  only  such  rays  as  are 
reflected  and  transmitted  in  circumstances  to  be  contiguous  to 
each  other,  and  to  continue  parallel  after  leaving  the  drop,  which 
can  produce  the  bright  colors  of  the  rainbow. 

640.  Course  of  Rays  in  the  Primary  Bow. — 

(Fig.  341)  be  the  section  of  a  drop  of 
rain,/>  a  diameter,  a~b,cd,  &c.,  par- 
allel rays  of  the  sun's  light,  falling 
upon  the  drop.  Now  yf9  a  ray  coin- 
ciding with  the  diameter,  suffers  no 
refraction;  and  a  b,  a  ray  near  to  y  f9 
is  refracted  very  little  toward  the  ra- 
dius, so  as  to  meet  the  remoter  surface 
of  the  drop  about  half  as  far  from  the 
axis  as  when  it  entered  ;  but  the  rays 
which  lie  further  from  yf9  making  greater  angles  with  the  radius, 
are  more  and  more  refracted  as  they  are  further  removed  from  the 
diameter. 


COURSE    OF    RAYS    IN    THE    TWO    BOWS.        383 

And  it  is  found,  by  a  simple  calculation  on  the  course  of  the 
rays,  that  those  which  enter  beyond  the  limit  of  about  60°,  cross 
more  or  less  of  those  entering  nearer  the  axis,  the  furthest  one  of 
all  at  90°  being  refracted  almost  to  p.  Hence  all  the  rays  falling 
on  the  quadrant/ 2,  meet  the  circumference  within  the  arc  kp. 
But  when  a  varying  quantity  is  approaching  its  limit,  or  is  begin- 
ning to  depart  from  it,  its  changes  are  nearly  insensible.  Thus,  a 
large  number  of  rays  near  c  d,  on  both  sides  of  it,  meet  very  near 
k,  the  limit  of  the  arc  p  k.  Consequently,  many  more  rays  are  re- 
flected from  that  point  than  from  any  other  in  the  arc.  Now 
were  these  rays  to  return  in  the  same  lines,  they  would  emerge 
parallel  in  the  lines  near  c  d ;  but  if,  instead  of  returning  back  in 
the  quadrant/ 2,  they  are  reflected  on  the  other  side  of  the  radius, 
they  make  the  same  angles  with  the  radius,  and  therefore  with 
each  other,  as  the  incident  rays  do,  and  consequently  meet  the 
curve  at  the  same  inclination  on  the  other  side  of  the  axis,  and 
emerge  parallel.  Hence  it  appears  that  there  is  a  particular  point 
in  the  section  of  the  drop  on  the  back  side,  where  the  rays  of  the 
sun's  light  accumulate,  and  then  diverge,  so  that,  on  emerging, 
those  of  a  given  color  form  a  compact  pencil  of  parallel  rays.  It 
is  found  by  calculation  that  the  angle  which  the  incident  and 
emergent  rays  make  with  each  other — that  is,  the  angle  included 
by  cd  and  eq  produced — is,  for  the  red  rays,  42°  2',  and  for  the 
violet  rays,  40°  17',  and  for  other  colors,  between  these  limits.  Cal- 
culation shows,  also,  that  these  are  the  greatest  deviations  possible 
for  rays  once  reflected;  since  all  rays  on  the  quadrant/ z,  whether 
nearer  or  further  than  the  pencil  c  d,  at  60°,  emerge  with  smaller 
deviations. 

641.  Course  of  Rays  in  the  Secondary  Bow.— There  is 
also  an  accumulation  at  a  certain  limit,  for  the  light  which 
emerges  after  suffering  two  reflections.  If,  as  before,  we  calculate 
the  course  of  the  rays  which  fall  on  the  quadrant/ 2,  we  shall  find 
that  those,  d  e  (Fig.  342),  which  enter  at  about  71°  or  72°,  from 
the  -axis  / p,  after  crossing  each  other 
in  the  drop  at  a,  are  reflected  at  h  into 
parallel  lines,  and,  consequently,  after 
a  second  reflection  at  m,  have  their  re- 
lations to  each  other  and  the  radii 
exactly  reversed.  Hence,  they  cross  a 
second  time  at  5,  and  emerge  parallel 
at  q.  Such  a  pencil,  entering  above  the 
axis,  will,  on  emerging,  ascend  and 
cross  its  own  path,  outside  of  the  drop,  the  violet  rays  intersecting 
d  e  at  an  angle  of  54°  9',  the  red  at  50°  59',  and  the  other  colors 


384  LIGHT. 

in  order  between.  That  the  emergent  pencil  may  descend  to  the 
observer,  the  incident  pencil  must  enter  fielotv  the  axis,  and  come 
out  above  it.  These  rays,  entering  at  the  distance  of  71°  and  72° 
from  the  axis,  are  the  only  ones  which,  after  tivo  reflections,  emerge 
compact  and  parallel,  and  give  a  bright  color  at  a  great  distance. 
All  rays  which  enter  nearer  the  axis,  and  also  those  which  enter 
more  remote,  make,  after  two  reflections,  larger  angles  of  devia- 
tion, and  also  diverge  from  each  other. 

642.  Axis  of  the  Bows.— Let  A  EDO  I  (Fig.  343)  repre- 
sent the  path  of  the  pencil  of  red  light  in  the  primary  bow.  If 
A  B  and  /  G  are  produced  to  meet  in  K,  the  angle  K  is  the  devi- 
ation, 42°  2',  of  the  incident  and  emergent  red  rays.  Suppose  the 
spectator  at  /,  and  let  a  line  from  the  sun  be  drawn  through  his 
position  to  T\  it  is  sensibly  parallel  to  A  B,  and  therefore  the 
angles  /  and  K  are  equal.  As  T  is  opposite  to  the  sun,  the  red 

FIG.  343.  FIG.  344. 


color  is  seen  at  the  distance  of  42°  2',  on  the  sky,  from  the  point  T\ 
and  so  the  angular  distance  of  each  color  from  T  equals  the  angle 
which  the  ray  of  that  color  makes  with  the  incident  ray.  In  like 
manner,  in  the  secondary  bow,  if  /  T  (Fig.  344)  be  drawn  through 
the  sun  and  the  eye  of  the  observer,  it  is  parallel  to  A  B,  and  the 
angular  distance  of  the  colored  ray  from  Tis  equal  to  K,  the  devi- 
ation of  the  incident  and  emergent  rays.  /  T  is  called  the  axis  of 
the  bows,  for  a  reason  which  is  explained  in  the  next  article. 

643.  Circular  Form  of  the  Bows.— Let  80  C  (Fig.  345) 
be  a  straight  line  passing  from  the  sun,  through  the  observer's 
place  at  0,  to  the  opposite  point  of  the  sky ;  and  let  V  0,  R:  0  be 
the  extreme  rays,  which  after  one  reflection  bring  colors  to  the  eye 
at  0,  and  R'  0,  V'O,  those  which  exhibit  colors  after  two  reflec- 
tions; then  (according  to  Arts.  640,  641),  VO  C~  40°  17',  ROC 
=  42°  2',  R1  0  C=  50°  59',  V'O  G=  54°  9'.  Now,  if  we  sup- 
pose the  whole  system  of  lines,  8  V'O,  8  V  0,  to  revolve  about 
8  0  C,  as  an  axis,  the  relations  of  the  rays  to  the  drops,  and  to 
each  other  will  not  be  at  all  changed ;  and  the  same  colors  will 
describe  the  same  lines,  whatever  positions  those  lines  may  occupy 


ORDER    OF    COLORS    IN    THE    TWO    BOWS.      385 

in  the  revolution.    The  emergent  rays,  therefore,  all  describe  the 
surfaces  of  cones,  whose  common  vertex  is  in  the  eye  at  0 ;  and 

FIG.  345. 


the  colors,  as  seen  on  the  cloud,  are  the  circumferences  of  their 
bases. 

In  a  given  position  of  the  observer,  the  extent  of  the  arches 
depends  on  the  elevation  of  the  sun.  When  on  the  horizon,  the 
bows  are  semicircles ;  but  less  as  the  sun  is  higher,  because  their 
centre  is  depressed  as  much  below  the  horizon  as  the  sun  is  ele- 
vated above  it.  If  rain  is  near,  however,  the  lower  parts  of  the 
bows  may  sometimes  be 'seen  projected*  on  the  landscape  as  arcs  of 
ellipses,  parabolas,  or  hyperbolas ;  for  the  surface  of  the  earth  cuts 
the  axis  of  the  cones  obliquely.  From  the  top  of  a  mountain,  the 
bows  have  been  seen  as  entire  circles. 

644.    Colors  of  the  Two  Bows  in  Reversed  Order.— 

The  reason  for  the  inversion  of  colors  in  the  two  bows  may  be 
seen  in  the  fact  that,  in  the  primary  bow,  the  rays  which  descend 
to  the  observer's  eye,  must  emerge  from  the  lower  or  inner  quad- 
rant of  the  drop,  and  be  bent  upward  (outward}  from  the  radius 
produced;  while,  in  the  secondary,  they  must  emerge  from  the 
upper  or  outer  quadrant,  and  be  bent  from  the  radius  downward. 
The  ray  V  0  (Fig.  345),  of  the  primary,  being  supposed  a,  violet 
ray,  is  the  most  refrangible,  and  therefore  all  other  rays  from  that 
drop  fall  below  it,  and  fail  to  reach  the  eye.  To  bring  other  colors 
to  0,  drops  must  be  selected  higher  up ;  hence,  violet  is  the  color 
seen  nearest  the  axis.  In  the  secondary  bow,  if  V  0  is  the  violet 
ray,  the  other  colors,  being  bent  in  a  less  degree  from  the  radius 
of  the  drop,  lie  above  V  0 ;  and  therefore,  in  order  that  other 
colors  may  reach  0,  they  must  emerge  from  lower  drops,  i.  e. 
drops  nearer  the  axis.  Hence,  violet  is  the  outer  color  of  the  sec- 
ondary bow. 

25 


386  LIGHT. 

645.  Rainbows,  the  Colored  Borders  of  Illuminated 
Segments  of  the  Sky. — The  primary  bow  is  to  be  regarded  as 
the  outer  edge  of  that  part  of  the  sky  from  which  rays  can  come  to 
the  eye  after  suffering  but  one  reflection  in  drops  of  rain ;  and  the 
secondary  bow  is  the  inner  edge  of  that  part  from  which  light, 
after  being  twice  reflected,  can  reach  the  eye. 

It  is  found  by  calculation,  that  in  case  of  one  reflection,  the 
incident  and  emergent  rays  can  make  no  inclinations  with  each 
other  greater  than  42°  2'  for  red  light,  and  40°  17'  for  violet;  but 
the  inclinations  may  be  less  in  any  degree  down  to  0°.  There- 
fore, all  light,  once  reflected,  comes  to  the  eye  from  within  the 
primary  bow. 

But  the  angles,  50°  59'  and  54°  9',  are,  by  calculation,  the  least 
deviations  of  red  and  violet  light  from  the  incident  rays  after  two 
reflections.  But  the  deviations  may  be  greater  than  these  limits 
up  to  180°.  Therefore  rays  twice  reflected  can  come  to  the  eye 
from  any  part  of  the  sky,  except  between  the  secondary  bow  and 
its  centre. 

It  appears,  then,  that  from  the  zone  lying  between  the  two 
bows,  no  light,  reflected  by  drops  internalty,  either  once  or  twice, 
can  possibly  reach  the  eye.  Observation  confirms  these  state- 
ments; when  the  bows  are  bright,  the  rain  within  the  primary  is 
more  luminous  than  elsewhere ;  and  outside  of  the  secondary  bow, 
there  is  more  illumination  than  between  the  two  bows,  where  the 
cloud  is  perceptibly  darkest. 

646.  The  Tertiary  Bow.— A  tertiary  bow,  or  a  bow  formed 
by  light  three  times  reflected  in  drops  of  rain,  is  on  the  same  side 
of  the  sky  with  the  sun,  and  distant  about  40°  40'  from  it.    The 
incident  rays,  which  form  it,  enter  the  drops  about  77°  from  their 
axis,  and  emerge  on  the  back  side.    But  this  order  of  bow  is  so 
very  faint  from  repeated  reflections,  and  so  unfavorably  situated, 
that  it  is  very  rarely  seen. 

647.  The  Common  Halo. — This,  as  usually  seen,  is  a  white 
or  colored  circle  of  about  22°  radius,  formed  around  the  sun  or 
moon.    It  might,  without  impropriety,  be  termed  the  frost-bow, 
since  it  is  known  to  be  formed  by  light  refracted  by  crystals  of  ice 
suspended  in  the  air.     It  is  formed  when  the  sun  or  moon  shines 
through  an  atmosphere  somewhat  hazy.    About  the  sun  it  is  a 
white  ring,  with  its  inner  edge  red,  and  somewhat  sharply  defined, 
while  its  outer  edge  is  colorless,  and  gradually  shades  off  into  the 
light  of  the  sky.    Around  the  moon  it  differs  only  in  showing  lit- 
tle or  no  color  on  the  inner  edge. 

648.  How  Caused. — The  phenomenon  is  produced  by  light 


THE    HALO. 


387 


passing  through  crystals  of  ice,  having  sides  inclined  to  each  other 
at  an  angle  of  60°.  Let  the  eye  be  at  E  (Fig.  346),  and  the  sun  in 
the  direction  E  S.  Let  S  A,  S  B,  &c.,  be  rays  striking  upon  sucli 
crystals  as  may  happen  to  lie  in  a  position  , 
to  refract  the  light  toward  8  E  as  an  axis. 
Each  crystal  turns  the  ray  from  the  re- 
fracting edge  on  entering ;  and  again,  on 
leaving,  it  is  bent  still  more,  and  the  emer- 
gent pencil  is  decomposed.  The  color,which 
comes  from  each  one  to  the  eye  E,  depends 
on  its  angular  distance  from  E  S,  and 
the  position  of  its  refracting  angle.  The 
angle  of  deviation  for  A  is  E  A  D=  S  E  A ; 
for  B,  it  is  S  E  B,  and  so  on.  It  is  found 
by  calculation,  that  the  least  deviation  for 
red  light  is  21°  45' ;  the  least  for  orange 
must  be  a  little  greater,  because  it  is  a  lit- 
tle more  refrangible,  and  so  on  for  the 
colors  in  order.  The  greatest  deviation 
for  the  rays  generally  is  about  43°  13'.  All 
light,  therefore,  which  can  be  transmitted 
by  such  crystals  must  come  to  the  observer  from  points  some- 
where between  these-two  limits,  21°  45'  and  43°  13'  from  the  sun. 
But  by  far  the  greater  part  of  it,  as  ascertained  by  calculation, 
passes  through  near  the  least  limit. 

649.  Its  Circular  Form. — What  takes  place  on  one  side  of 
E  8  may  occur  on  every  side ;  or,  in  other  words,  we  may  suppose 
the  figure  revolved  about  E  8  as  an  axis,  and  then  the  transmitted 
light  will  appear  in  a  ring  about  the  sun  S.    The  inner  edge  of 
the  ring  is  red,  since  that  color  deviates  least;  just  outside  of  the 
red  the  orange  mingles  with  it;  beyond  that  are  the  red,  orange, 
and  yellow  combined ;  and  so  on,  till,  at  the  minimum  angle  for 
violet,  all  the  colors  will  exist  (though  not  in  equal  proportions), 
and  the  violet  will  be  scarcely  distinguishable  from  white.    Beyond 
this  narrow 'colored  band  the  halo  is  white,  growing  more  and 
more  faint,  so  that  its  outer  limit  is  not  discernible  at  all. 

650.  The  Halo,  a  Bright  Border  of  an  Illuminated  Zone. 

— As  in  the  rainbow,  so  in  the  halo,  the  visible  band  of  colors  is 
only  the  border  of  a  large  illuminated  space  on  the  sky.  The 
ordinary  halo,  therefore,  is  the  bright  inner  border  of  a  zone,  which 
is  more  than  20°  wide.  The  whole  zone,  except  the  inner  edge,  is 
too  faint  to  be  generally  noticed,  though  it  is  perceptibly  more 
luminous  than  the  space  between  the  halo  and  the  luminary. 


388  LIGHT. 

651.  Frequency  of  the  Halo. — The  halo  is  less  brilliant 
and  beautiful,  but  far  more  frequent,  than  the  rainbow.     Scarcely 
a  week  passes  during  the  whole  year  in  which  the  phenomenon 
does  not  occur.     In  summer  the  crystals  are  three  or  four  miles 
high,  above  the  limit  of  perpetual  frost.    As  the  rainbow  is  some- 
times seen  in  dew-drops  on  the  ground,  so  the  frost-bow,  just  after 
sunrise,  has  been  noticed  in  the  crystals  which  fringe  the  grass. 

652.  The   Mock   Sun. — The   mock  sun,  or  sun-dog,  is  a 
short  arc  of  the  halo,  occasionally  seen  at  22°  distance,  on  the 
right  and  left  of  the  sun,  when  near  the  horizon.    The  crystals, 
which  are  concerned  in  producing  the  mock  sun,  are  supposed  to 
have  the  form  of  spiculce,  or  six-sided  needles,  whose  alternate  sides 
are  inclined  to  each  other  at  an  angle  of  60° ;  these,  if  suspended 
in  the  air  in  a  vertical  position,  could  refract  the  light  only  in 
directions  nearly  horizontal,  and  therefore  present  only  the  right 
and  left  sides  of  the  halo. 

In  high  latitudes,  other  and  complex  forms  of  halo  are  fre- 
quent, depending  for  their  formation  on  the  prevalence  of  crystals 
of  other  angles  than  60°.  [See  Appendix  for  calculations  of  the 
angular  radius  of  rainbows  and  halo.] 


CHAPTER    VI. 

COLOR,  BY  REFLECTION,  INFLECTION,  STRIATION  OF  SURFACE, 
AND  THIN  PLATES. 

653.  Natural  Colors  of  Bodies. — The  colors  which 'bodies 
exhibit,  when  seen  in  ordinary  white  light,  are  owing  to  the  fact 
that  they  decompose  light  by  absorbing  or  transmitting  some 
colors  and  reflecting  the  others.  We  say  that  a  body  has  a  certain 
color,  whereas  it  only  reflects  that  color ;  a  flower  is  called  red, 
because  it  reflects  only  or  principally  red  light;  another  yellow, 
because  it  reflects  yellow  light,  &c.  A  white  surface  is  one  which 
reflects  all  colors  in  their  due  proportion ;  and  such  a  surface, 
placed  in  the  spectrum,  assumes  each  color  perfectly,  since  it  is 
capable  of  reflecting  all.  A  substance  which  reflects  no  light,  or 
but  very  little,  is  black.  What  peculiarity  of  constitution  that  is 
which  causes  a  substance  to  reflect  a  certain  color,  and  to  absorb 
others,  is  unknown. 

Very  few  objects  have  a  color  which  exactly  corresponds  to  any 
color  of  the  spectrum.  This  is  found  to  result  from  the  fact  that 


INFLECTION    OF    LIGHT. 

most  bodies,  while  they  reflect  some  one  color  chiefly,  reflect  the 
others  in  some  degree.  A  red  flower  reflects  the  red  light  abun- 
dantly, and  perhaps  some  rays  of  all  the  other  colors  with  the  red. 
Hence  there  may  be  as  many  shades  of  red  as  there  can  be  differ- 
ent proportions  of  other  colors  intermingled  with  it.  The  same 
is  true  of  each  color  of  the  spectrum.  Thus  there  is  an  infinite 
variety  of  tints  in  natural  objects.  .  These  facts  are  readily  estab- 
lished by  using  the  prism  to  decompose  the  light  which  bodies 
reflect. 

654.  Inflection,  or  Diffraction  of  Light. — This  phenome- 
non consists  of  delicate  colored  fringes  bordering  the  edges  of  shad- 
ows, when  the  light  comes  from  a  luminous  point  or  line. 

For  the  purpose  of  experiments  on  this  subject,  a  beam  of  light 
is  admitted  into  a  dark  room,  through  a  very  small  aperture,  as  a 
pin-hole  made  in  sheet-lead ;  or,  what  is  better,  a  convex  lens  is 
placed  in  the  window-shutter,  which  brings  the  rays  to  a  focus, 
and  affords  a  divergent  pencil  of  light.  If  we  introduce  into  this 
pencil  any  opaque  body,  as  a  knife-blade,  for  example,  and  observe 
the  shadow  which  it  casts  on  a  white  screen,  we  shall  observe  on 
both  sides  of  the  shadow  fringes  of  colored  light,  the  different  col- 
ors succeeding  each  other  in  the  order  of  the  spectrum,  from  violet 
to  red.  Three  or  four  series  can  usually  be  discerned,  the  one 
nearest  to  the  shadow  being  the  most  complete  and  distinct,  and 
the  remoter  ones  having  fewer  and  fainter  colors.  The  phenome- 
non is  independent  of  the  density  or  thickness  of  the  body  which 
casts  the  shadow.  The  light,  in  passing  by  the  edge  or  back  of  a 
knife,  by  a  block  of  marble  or  a  bubble  of  air  in  glass,  is  in  each 
case  affected  in  the  same  way.  But  if  the  body  is  very  narrow,  as, 
for  example,  a  fine  wire,  a  modification  arises  from  the  light  which 
passes  the  opposite  side ;  for  now  fringes  appear  within  the  shadow, 
and  at  a  certain  distance  of  the  screen  the  central  line  of  the 
shadow  is  the  most  luminous  part  of  it. 

655.  Breadth  of  Fringe  varies  with  the  Color. — If,  in 

the  foregoing  experiments,  we  use  light  of  one  color  alone  instead 
of  white  light,  then  the  fringes  are  only  of  that  color,  separated 
from  each  other  by  lines  which  are  comparatively  dark ;  and,  on 
measuring  the  breadths  and  distances  of  fringes  of  different  colors, 
those  of  red  light  are  found  to  be  widest,  those  of  violet  narrowest, 
and  the  other  colors  have  breadths  according  to  their  order.  This 
explains  why,  in  the  case  of  white  light,  the  several  colors  appear 
in  a  series,  with  the  red  outermost ;  for  each  element  of  the  white 
light  forms  its  own  system  of  fringes,  but  the  systems  do  not  coin- 
cide— the  wider  ones  project  beyond  the  narrower,  and  thus  be- 
come separately  visible. 


390  LIGHT. 

If  the  screen  is  moved  further  from  the  body,  the  distance  of  a 
given  color  from  the  edge  of  the  shadow  becomes  greater,  but  not 
in  proportion  to  the  distance  of  the  screen  from  the  body;  which 
proves  that  the  color  is  not  propagated  in  a  straight  line,  but  in  a 
curve.  These  curves  are  found  to  be  hyperbolas,  having  their  con- 
cavity on  the  side  next  the  shadow,  and  are  in  fact  a  species  of 
caustics. 

656.  Light  through  Small  Apertures. — The  phenomena  of 
inflection  are  exhibited  in  a  more  interesting  manner  when  we 
view  with  a  magnifying  glass  a  pencil  of  light  after  it  has  passed 
through  a  small  aperture.     For  instance,  in  the  cone  already  de- 
scribed as  radiating  from  the  focus  of  a  lens  in  a  dark  room,  let  a 
plate  of  lead  be  interposed,  having  a  pin-hole  pierced  through  it, 
and  let  the  slender  pencil  of  light  which  passes  through  the  pin- 
hole  fall  on  the  magnifier.    The  aperture  will  be  seen  as  a  lumi- 
nous circle  surrounded  by  several  rings,  each  consisting  of  a  pris- 
matic series.    These  are,  in  truth,  the  fringes  formed  by  the  edge 
of  the  circular  puncture,  but  they  are  modified  by  the  circum- 
stance that  the  opposite  edges  are  so  near  to  each  other.    If,  now, 
the  plate  be  removed,  and  another  interposed  having  two  pin-holes, 
within  one-eighth  of  an  inch  of  each  other,  besides  the  colored 
rings  round  each,  there  is  the  additional  phenomenon  of  long  lines 
crossing  the  space  between  the  apertures;  the  lines  are  nearly 
straight,  and  alternately  luminous  and  dark,  and  varying  in  color, 
according  to  their  distance  from  the  central  one.     These  lines  are 
wholly  due  to  the  overlapping  of  two  pencils  of  light,  for  on  cov- 
ering one  of  the  apertures  they  entirely  disappear.     By  combining 
circular  apertures  and  narrow  slits  in  various  patterns  in  the  screen 
of  lead,  very  brilliant  and  beautiful  effects  are  produced. 

657.  Why  Inflection  is  not  always  noticed  in  looking 
by  the  Edges  of  Bodies. — It  must  be  understood  that  light  is 
always  inflected  when  it  passes  by  the  edges  of  bodies ;  but  that  it 
is  rarely  observed,  because,  as  light  comes  from  various  sources  at 
once,  the  colors  of  each  pencil  are  overlapped  and  reduced  to 
whiteness  by  those  of  all  the  others.     By  using  care  to  admit  into 
the  eye  only  isolated  pencils  of  light,  some  cases  of  inflection  may 
be  observed  which  require  no  apparatus.     If  a  person  standing  at 
some  distance  from  a  window  holds  close  to  his  eye  a  book  or 
other  object  having  a  straight  edge,  and  passes  it  along  so  as  to 
come  into  apparent  coincidence  with  the  sash-bars  of  the  window, 
he  will  notice,  when  the  edge  of  the  book  and  the  bar  are  very 
nearly  in  a  range,  that  the  latter  is  bordered  with  colors,  the  violet 
extremity  of  the  spectrum  being  on  the  side  of  the  bar  nearest  to 


COLOR    BY    THIN    LAMINA.  391 

the  book,  and  the  red  extremity  on  the  other  side.  Again,  the 
effect  produced  when  light  passes  through  a  narrow  aperture  may 
be  seen  by  looking  at  a  distant  lamp  through  the  space  between 
the  bars  of  a  pocket-rule,  or  between  any  two  straight  edges 
brought  almost  into  contact.  On  each  side  of  the  lamp  are  seen 
several  images  of  it,  growing  fainter  with  increased  distance,  and 
finely  colored.  An  experiment  still  more  interesting  is  to  look  at 
a  distant  lamp  through  the  net-work  of  a  bird's  feather.  There 
are  several  series  of  colored  images,  having  a  fixed  arrangement  in 
relation  to  the  disposition  of  the  minute  apertures  in  the  feather ; 
for  the  system  of  images  revolves  just  as  the  feather  itself  is 
revolved. 

658.  Striated  Surfaces. — If.  the  surface  of  any  substance  is 
ruled  with  fine  parallel  grooves,  2000  or  more  to  the  inch,  it  will 
reflect  bright  colors  when  placed  in  the  sunbeam.    Mother-of- 
pearl  and  many  kinds  of  sea-shell  exhibit  colors  on  account  of 
delicate  striae  on  their  surface.    These  are  the  edges  of  thin  lam- 
ina3  which  compose  the  shell,  and  which  crop  out  on  the  surface 
in  fine  and  nearly  parallel  lines.    It  may  be  known  that  the  color 
arises  from  such  a  cause,  if,  when  the  substance  is  impressed  on 
fine  cement,  its  colors  are  communicated  to  the  cement.    Indeed, 
it  was  in  this  way  that  Dr.  Wollaston  accidentally  discovered  the 
true  cause  of  such  colors.    The  changeable  hues  in  the  plumage 
of  some  birds,  and.  the  wings  of  some  insects,  are  owing  to  a 
striated  structure  of  their  surfaces.     But  the  metals  can  be  made 
to  furnish  the  most  brilliant  spectra,  by  stamping  them  with  steel 
dies,  which  have  been  first  ruled  by  a  diamond  with  lines  from 
2000  to  10,000  per  inch,  and  then  hardened.    Gilt  buttons  and 
other  articles  for  dress  are  sometimes  prepared  in  this  manner, 
and  are  called  iris  ornaments.    The  color  in  a  given  case  depends 
on  the  distance  between  the  grooves,  and  the  obliquity  of  the  beam 
of  light.    Hence,  the  same  surface,  uniformly  striated,  may  reflect 
all  the  colors,  and  every  color  many  times,  by  a  mere  change  in  its 
inclination  to  the  beam  of  light. 

659.  Thin  Laminae. — Any  transparent  substance,  when  re- 
duced in  thickness  to  a  few  millionths  of  an  inch,  reflects  brilliant 
colors,  which  vary  with  every  change  of  thickness.    Examples  are 
seen  in  the  thin  laminae  of  air  occupying  cracks  in  glass  and  ice, 
and  the  interstices  between  plates  of  mica,  also  in  thin  films  of  oil 
on  water,  and  alcohol  on  glass,  but  most  remarkably  in  soapy 
water  blown  into  very  thin  bubbles. 

If  a  lens  of  slight  convexity  is  laid  on  a  plane  lens,  and  the 
two  are  pressed  together  by  a  screw,  and  viewed  by  reflected  light, 


392  LIGHT. 

rings  of  color  are  seen  arranged  around  the  point  of  contact.  The 
rings  of  least  diameter  are  broadest  and  most  brilliant,  and  each 
one  contains  the  colors  of  the  spectrum  in  their  order,  from  violet 
on  the  inner  edge  to  red  on  the  outer.  But  the  larger  rings  not 
only  become  narrower  and  paler,  but  contain  fewer  colors ;  yet  the 
succession  is  always  in  the  same  order  as  above.  Increased  pres- 
sure causes  the  rings  to  dilate,  while  new  ones  start  up  at  the 
centre,  and  enlarge  also,  until  the  centre  becomes  black,  after 
which  no  new  rings  are  formed.  These  are  commonly  called 
Newton's  rings,  because  Sir  Isaac  Newton  first  investigated  their 
phenomena. 

660.  Ratio  of  Thicknesses  for  Successive  Rings. — A 
given  color  appears  in  a  circle  around  the  point  of  contact,  be- 
cause equal  thicknesses  are  thus  arranged.    If  the  diameters  of 
the  successive  rings  of  any  one  color  be  carefully  measured,  their 
squares  are  found  to  be  as  the  odd  numbers,  1,  3,  5,  7 ;  and  hence 
the  thicknesses  of  the  laminae  of  air  at  the  repetitions  of  the  same 
color  are  as  the  same  numbers.    For,  let  Fig.  347  represent  a  sec- 
tion of  the  spherical  and  plane 

surfaces  in  contact  at  a.    Let  FrG-  347- 

a b,  ad,  be  the  radii  of  two 

rings  at  their  brightest  points. 

Suppose  a  i,  perpendicular  to 

m  n,  to   be  produced   till  it 

meets   the  opposite  point  of 

the  circle  of  which  ag  is  an  arc,  and  call  that  point/;  then  af  is 

the  diameter  of  the  sphere  of  which  the  lens  is  a  segment.    Let 

be,  dg,  be  parallel  to  a  i,  and  e  li,  g  i,  to  m  n,  then  we  have 

(e7iY  :  (giY  : :  ah  x  hf:ai  x  if. 

But  the  distances  between  the  two  lenses  being  exceedingly 
small  in  comparison  with  the  diameter  of  the  sphere,  hf  and  if 
may  be  taken  as  equal  to  af,  whence, 'by  substitution, 

(e  h)* :  (gi)* ;  iah  x  af\ai  x  af : :  a  Ji :  a  i  -  :  1)  e  :  dg. 

Therefore  the  thicknesses  of  successive  rings  are  as  the  odd 
numbers. 

661.  Thickness  of  Laminae  for  Newton's  Rings. — The 

absolute  thickness,  be,dg,  &c.,  can  also  be  obtained,  af  being 
known,  since 

af :  ae  ::  ae  :  ah  OTfie; 

for  in  so  short  arcs  the  chord  may  be  considered  equal  to  the  sine, 
that  is,  the  radius  of  the  ring.  When  air  is  between  the  lenses, 
all  the  rings  range  between  the  thickness  of  lialf  a  millionth  of  an 


DOUBLE  REFRACTION.  393 

inch  and  72  millionth*  j  if  water  is  used,  the  limits  are  J  of  a  mil- 
lionth and  58  millionths.  Below  the  smaller  limit  the  medium 
appears  black,  or  no  color  is  reflected ;  above  the  highest  limit 
the  medium  appears  white,  all  colors  being  reflected  together. 
When  water  is  substituted  for  air,  all  the  rings  contract  in  diam- 
eter, indicating  that  a  particular  order  of  color  requires  less  thick- 
ness of  water  than  of  air ;  the  thicknesses  for  different  media  are 
found  to  be  in  the  inverse  ratio  of  the  indices  of  refraction. 

662.  Relation  of  Rings  by  Reflection  and  by  Transmis- 
sion.— If  the  eye  is  placed  beyond  the  lenses,  the  transmitted  light 
also  is  seen  to  be  arranged  in  very  faint  rings,  the  brightest  por- 
tions being  at  the  same  thicknesses  as  the  darkest  ones  by  reflec- 
tion ;  and  these  thicknesses  are  as  the  even  numbers,  2,  4,  6,  &c. 
The  centre,  when  black  by  reflection,  is  white  by  transmission, 
and  where  red  appears  on  one  side,  blue  is  seen  on  the  other; 
and,  in  like  manner,  each  color  by  reflection  answers  to  its  comple- 
mentary color  by  transmission. 

663.  Newton's  Rings  by  a  Monochromatic  Lamp. — The 

number  of  reflected  rings  seen  in  common  light  is  not  usually 
greater  than  from  Jive  to  ten.  The  number  is  thus  small,  because 
as  the  outer  rings  grow  narrower  by  a  more  rapid  separation  of 
the  surfaces,  the  different  colors  overlap  each  other,  and  produce 
whiteness.  But  if  a  light  of  only  one  color  falls  on  the  lenses,  the 
number  may  be  multiplied  to  several  hundreds ;  the  rings  are 
alternately  of  that  color  and  black,  growing  more  and  more  nar- 
row at  greater  distances,  till  they  can  be  traced  only  by  a  micro- 
scope. A  good  light  for  such  a  purpose  is  the  flame  of  an  alcohol 
lamp,  whose  wick  has  been  soaked  in  strong  brine,  and  dried. 


CHAPTER    VII. 

DOUBLE  REFRACTION  AND  POLARIZATION. 

664.  Double  Refraction. — There  are  many  transparent  sub- 
stances, particularly  those  of  a  crystalline  structure,  which,  instead 
of  refracting  a  beam  of  light  in  the  ordinary  mode,  divide  it  into 
two  beams.  This  effect  is  called  double  refraction,  and  substances 
which  produce  it  are  called  doubly-refracting  substances. 

This  phenomenon  was  first  observed  in  a  crystal  of  carbonate 
of  lime,  denominated  Iceland  spar.  It  is  bounded  by  six  rhom- 


394 


LIGHT. 


FIG.  348. 


FIG.  349. 


boidal  faces,  whose  inclinations  to  each  other  are  either  105°  5',  or 
74°  55'.  There  are  two  opposite  solid  angles,  A  and  X  (Fig.  348), 
each  of  which  is  formed  by  the  meeting  of  three 
obtuse  plane  angles ;  and  when  the  edges  of  the 
crystal  are  equal,,  the  diagonal  A  X  is  equally 
inclined  to  the  edges  which  it  meets,  as  A  B, 
A  C,  and  AD',  A  X  is  called  the  axis  of  the 
crystal.  But  every  other  line  in  the  crystal  par- 
allel to  A  JTis  also  an  axis,  because  the  crystal 
may  be  conceived  to  be  divided  into  any  number 
of  similar  crystals,  each  having  its  own  axis ;  the  axis  is  therefore 
a  direction  rather  than  a  line.  If  a  thick  crystal  of  spar  be  laid  on 
a  line  of  writing,  it  appears  as  two  lines,  one  of  which  seems  not 
only  thrown  aside  from  the  other,  but  brought  a  little  nearer  to 
the  eye.  Therefore  every  ray  of  light,  in  passing  through,  is  di- 
vided into  two  rays,  which  come  to  the  eye  in  different  directions. 
The  double  refraction  may  also  be  seen  by  letting  a  very  slender 
sunbeam,  R  r  (Fig.  349),  fall  on  the  crystal ;  as  it  enters  it  takes 
two  directions,  r  0,  and  r  E,  which  on  passing 
out  describe  the  lines  0  0',  E-  E',  parallel  to 
the  incident  beam,  R  r.  One  of  these  rays, 
0  0')  is  called  the  ordinary  ray,  because  it  is 
always  refracted  according  to  the  ordinary  law 
of  refraction  (Art.  606) ;  that  is,  it  remains  in 
the  plane  of  incidence,  and  the  sines  of  inci- 
dence and  refraction  have  a  constant  ratio  to 
each  other  at  all  inclinations.  The  other,  E E', 
is  called  the  extraordinary  ray,  because  in  some 
positions  it  departs  from  this  law  of  refraction 
in  one  or  both  particulars. 

The  property  of  double  refraction  belongs  to  a  large  num- 
ber of  crystals,  and  also  to  some  animal  substances,  as  hair, 
quills,  &c. ;  and  it  may  be  produced  artificially  in  glass  by  heat  or 
pressure. 

665.  Optical  Relations  of  the  Axis.— The  axis  of  Iceland 
spar  has  been  defined  with  reference  to  form  ;  but  it  is  also  the 
axis  with  respect  to  its  optical  relations,  for  in  the  direction  of 
that  line  a  ray  is  never  doubly  refracted,  while  it  is  doubly  re- 
fracted in  all  other  directions. 

Every  plane  which  includes  the  axis  of  a  crystal  is  called  a 
principal  section.  In  every  principal  section  the  extraordinary 
ray  conforms  to  one  part  of  the  law  of  refraction,  but  not  to  the 
other ;  it  remains  in  the  plane  of  incidence,  but  does  not  preserve 
a  constant  ratio  of  sines  at  different  inclinations. 


POLARIZATION    BY    REFLECTION.  395 

In  a  plane  at  right  angles  to  the  axis,  the  extraordinary  ray 
conforms  to  both  parts  of  the  law  ;  but  in  all  planes  besides  this 
and  the  principal  sections,  it  conforms  to  neither  part. 

Crystals  of  a  positive  axis,  are  those  in  which  the  extraordinary 
ray  has  a  larger  index  of  refraction  than  the  ordinary  ray  ;  crys- 
tals of  a  negative  axis  are  those  in  which  the  index  of  the  extraor- 
dinary ray  is  less  than  that  of  the  ordinary  ray.  Iceland  spar  is  a 
crystal  of  negative  axis. 

Some  crystals  have  two  axes  of  double  refraction;  that  is, 
there  are  two  directions  in  which  light  may  be  transmitted  with- 
out being  doubly  refracted.  A  few  crystals  have  more  than  two 
axes. 

666.  Polarization  of  Light.  —  This  name  is  given  to  a 
change  which  may  be  produced  in  light,  such  that  it  has  different 
properties  on  different  sides.  Common  light,  as,  for  instance,  a  di- 
rect sunbeam,  has  the  same  relation  to  space  on  all  sides.  If  it 
falls  on  a  piece  of  glass  at  a  given  angle,  it  will  suffer  reflection 
equally  well  in  every  plane,  as  we  turn  the  glass  round,  and  so  of 
refraction,  or  any  change  we  may  attempt.  But  if  a  beam  were 
so  changed  in  its  character  that  it  could  be  reflected  upward,  but 
could  not  be  reflected  to  the  right,  it  would  be  called,  not  common, 
but  polarized  light. 


667.  Polarization  by  Reflection.  —  Let  two  tubes, 
N  P  (Fig.  350),  be  fitted  together  in  such  a  manner  that  one  can 

FIG.  350. 


be  revolved  upon  the  other ;  and  to  the  end  of  each  let  there  be 
attached  a  plate  of  dark-colored  glass,  A  and  O,  capable  of  reflect- 
ing only  from  the  first  surface.  These  plates  are  hinged  so  as  to 
be  adjusted  at  any  angle  with  the  axis  of  the  tube.  Let  the  plane 
of  each  glass  incline  to  the  axis  of  the  tube  at  an  angle  of  33°, 
and  let  the  beam  R  A  make  an  incidence  of  57°,  the  complement 


396  LIGHT. 

of  33°,  on  A  ;  then  it  will,  after  reflection,  pass  along  the  axis  of 
the  tube,  and  make  the  same  angle  of  incidence  on  C.  If  now 
the  tube  NP  be  revolved,  the  second  reflected  ray  will  vary  its 
intensity,  according  to  the  angle  between  the  two  planes  of  inci- 
dence on  A  and  C.  The  beam  A  C  is  polarized  light ;  the  glass 
A,  which  has  produced  the  polarization,  is  called  the  polarizing 
plate  ;  the  glass  C,  which  shows,  by  the  effects  of  its  revolution, 
that  A  C  is  polarized,  is  the  analyzing  plate;  and  the  whole  in- 
strument, constructed  as  here  represented,  or  in  any  other  manner 
for  the  same  purpose,  is  called  &  polariscope. 

668.  Changes  of  Intensity  Described. — The  changes  in 
the  ray  C E  are  as  follows :  When  the  tube  NP  is  placed  so  that 
the  plane  of  incidence  on  C  is  coincident  with  the  former  plane 
of  incidence,  R  A  C,  whether  C  E  is  reflected  forward  or  back- 
ward in  that  plane,  the  intensity  at  E  will  be  the  same  as  if  A  C 
had  been  a  beam  of  common  light.    If  N  P  is  revolved,  E  will 
begin  to  grow  fainter,  and  reach  its  minimum  of  intensity  when 
the  planes  RAG  and  A  CE  are  at  right  angles,  which  is  the 
position  indicated  in  the  figure.    Continuing  the  revolution,  we 
find  the  intensity  increasing  through  the  second  quadrant  of  rev- 
olution, and  reaching  its  maximum,  when  the  two  planes  of  inci- 
dence again  coincide,  180°  from  the  first  position.    The  next  half 
revolution  repeats  these  changes  in  the  same  order. 

669.  The  Polarizing  Angle.— The  angle  of  57°  is  called  the 
polarizing  angle  for  glass,  not  because  glass  will  not  polarize  at 
other  angles  of  incidence,  but  because  at  all  other  angles  it  po- 
larizes the  light  in  a  less  degree ;  and  this  is  indicated  by  the  fact 
that,  in  revolving  the  analyzing  plate,  there  is  less  change  of  in- 
tansity,  and  the  light  at  E  does  not  become  so  faint.    Different 
substances  have  different  polarizing  angles,  and  these  are  found  to 
be  so  connected  with  the  degree  of  refractive  power,  that  by  a 
knowledge  of  the  index  of  refraction  for  any  substance,  its  polar- 
izing angle  can  be  calculated,  and  vice  versa.    Hence  the  refractive 
power  of  opaque  bodies  may  be  determined.     No  substance  en- 
tirely polarizes  the  light  incident  upon  it,  even  at  the  angle  of 
polarization.      Complete   polarization  of  the  ray  A  C  would  be 
indicated  by  the  entire  extinction  of  C  E,  at  two  opposite  points 
of  its  revolution.     On  the  other  hand,  every  substance  polarizes, 
in  some  degree,  the  light  which  it  reflects.    The  polarization  pro- 
duced by  reflection  from  the  metals  is  very  slight. 

670.  Polarization  by  a  Bundle  of  Plates. — Light  may 
also  be  polarized  by  transmission  through  a  bundle  of  laminae  of 


EVERY    POLARIZER    AN    ANALYZER.  397 

a  transparent  substance,  at  an  angle  of  incidence  equal  to  its  po- 
larizing angle.  Let  a  pile  of  twenty  or  thirty  plates  of  transpa- 
rent glass,  no  matter  how  thin,  be  placed  in  the  same  position  as 
the  reflector  A,  in  Fig.  350,  and  a  beam  of  light  be  transmitted 
through  them  in  a  direction  toward  C.  In  entering  and  leaving 
the  bundle  A,  situated  as  in  the  figure,  the  angles  of  incidence 
and  refraction  are  in  a  horizontal  plane.  When  C  is  revolved,  the 
beam  undergoes  the  same  changes  as  before,  with  this  difference, 
that  the  places  of  greatest  and  least  intensity  will  be  reversed.  If 
the  light  is  reflected  from  C  in  the  same  plane  in  which  it  was  re- 
fracted by  A^  its  intensity  is  least,  and  it  is  greatest  when  reflected 
in  a  plane  at  right  angles  to  it,  as  at  E  in  the  figure. 

671.  Polarization  by  Crystals. — The  third  and  most  per- 
fect method  of  polarizing  light,  is  by  transmission  through  certain 
crystals.    Some  crystals  polarize  the  transmitted  light  by  absorp- 
tion ;  and  every  doubly-refracting  crystal  polarizes  both  the  ordi- 
nary and  the  extraordinary  ray.    If  a  thin  plate  be  cut  from  a 
crystal  of  tourmaline,  by  planes  parallel  to   its  axis,  the  beam 
transmitted  through  it  is  polarized,  and,  when  received  on  the 
analyzing  plate,  will  alternately  become  bright  and  faint,  as  the 
tube  of  the  analyzer  is  revolved.    And  if  a  beam  is  passed  through 
a  deubly-refracting  crystal,  and  the  two  parts  fall  on  the  analyzing 
plate,  they  will  come  to  their  points  of  greatest  and  least  bright- 
ness at  alternate  quadrants ;   indeed,  when  one  ray  is  brightest, 
the  other  is  entirely  extinguished.     Therefore  the  two  rays  which 
emerge  from  a  doubly-refracting  crystal  are  polarized  completely, 
and  in  planes  at  right  angles  with  each  other. 

672.  Every  Polarizer  an  Analyzer. — We  have  seen  that 
light  is  polarized  by  reflection  from  glass  at  an  incidence  of  57°, 
and  analyzed  by  another  plate  at  the  same  angle  of  incidence. 
This  is  but  an  instance  of  what  is  always  true,  that  every  method 
of  polarizing  light  may  be  used  to  analyze,  i.  e.,  to  test  its  polar- 
ization.   Hence,  a  bundle  of  thin  plates  of  glass  may  take  the 
place  of  the  analyzer  (7,  as  well  as  of  the  polarizer  A.     For,  on 
turning  it  round,  though  the  transmitted  beam  remains  in  the 
same  place,  yet  it  will,  at  the  alternate  quadrants,  brighten  to  its 
maximum  and  fade  to  its  minimum  of  intensity. 

So,  again,  if  light  has  passed  through  a  tourmaline,  and  is 
received  on  a  second  whose  crystalline  axis  is  parallel  to  that  of 
the  former,  the  ray  will  proceed  through  that  also ;  but  if  the 
second  is  turned  in  its  own  plane,  the  transmitted  ray  grows  faint, 
and  nearly  disappears  at  the  moment  when  the  two  axes  are  at 


398  LIGHT. 

90°  of  inclination,  and  this  alternation  continues  at  each  90°  of 
the  whole  revolution. 

Finally,  place  a  double-refractor  at  each  end  of  the  polariscope, 
and  let  a  beam  pass  through  them  and  fall  on  a  screen.  The  first 
crystal  will  polarize  each  ray,  and  the  second  will  doubly  refract 
and  also  analyze  each,  exhibiting  a  very  interesting  series  of 
changes.  In  general,  four  rays  will  emerge  from  the  second  crys- 
tal, producing  four  luminous  spots  on  the  screen.  But,  on  re- 
volving the  tube,  not  only  do  the  rays  commence  a  revolution 
round  each  other,  but  two  of  them  increase  in  brightness,  and  the 
other  two  at  the  same  time  diminish  as  fast,  till  two  alone  are  vis- 
ible, at  their  greatest  intensity.  At  the  end  of  the  second  quad- 
rant, the  spots  before  invisible  are  at  their  maximum  of  bright- 
ness, and  the  others  are  extinguished.  This  alternation  continues 
as  long  as  the  crystal  is  revolved.  In  the  middle  of  each  quadrant 
the  four  are  of  equal  brightness. 

673.  Color  by  Polarized  Light.— The  phenomena  of  color 
produced  by  polarized  light  are  beautiful,  and  of  great  interest. 

Let  a  very  thin  plate  of  some  doubly-refracting  crystal  be 
placed  perpendicularly  across  the  axis  of  the  polariscope  (Fig.  351), 
and  let  the  analyzed  ray,  C  E,  fall  on  a  screen.    When  the  princi- 
pal section  of  the  crystal,  D  Hy 
coincides  with  the  first  plane  FlG-  351. 

of  reflection,  R  A  C,  or  is  per- 
pendicular to  it,  all  the  phe- 
nomena are  the  same  as  if  no 
crystal  was  interposed.  But 
let  the  film  be  revolved  in  its 
own  plane  till  D  H  makes  45° 
with  the  plane  RAG',  then, 

instead  of  the  dark  spot  at  E,  a  brilliant  color  appears.  That 
color  may  be  any  tint  of  the  spectrum,  according  to  the  thickness 
of  the  interposed  film.  If  now  the  revolution  of  the  crystal  is 
continued,  the  color  fades  out  at  the  end  of  the  next  45°,  reap- 
pears at  90°,  and  so  on.  But  if  the  crystal  be  so  placed  as  to  give 
color,  and  the  analyzing  plate  be  revolved,  a  different  series  pre- 
sents itself.  The  color  observed  at  E,  during  the  first  45°,  grad- 
ually fades,  and  during  the  next  45°  its  complement  appears  and 
brightens  to  its  maximum.  The  original  color  is  restored  at  180°, 
and  the  complementary  color  at  270°. 

The  most  interesting  form  of  this  experiment  is  seen  when  the 
light  is  polarized  and  analyzed  by  means  of  double-refractors; 
since  the  polarization  is  more  perfect,  and  the  two  pairs  of  oppo- 
sitely polarized  rays  are  on  the  screen  at  once.  "When  two  of  the 


SYSTEMS    OF    COLORED    RINGS.  399 

images  are  of  a  certain  color,  the  other  two  have  the  complemen- 
tary color. 

674.  Systems  of  Colored  Rings. — Systems  of  irised  Lands 
and  rings  may  also  be  produced  by  the  polariscope.  Let  a  plate 
be  cut  from  a  doubly-refracting  crystal  of  one  axis  by  planes  per- 
pendicular to  that  axis;  and  place  it  between  the  polarizer  arid 
analyzer.  If  now  a  pencil  of  sufficient  divergency  is  transmitted, 
a  system  of  colored  circles  will  be  formed,  resembling  Newton's 
rings  between  lenses.  If  a  polariscope  is  formed  of  two  tourma- 
lines, and  the  crystal  laid  between  them,  and  the  whole  combina- 
tion, less  than  half  an  inch  thick,  is  brought  close  to  the  eye,  the 
pencil  of  light  will  consist  of  rays  of  various  obliquity,  and  the 
rings  may  be  seen  beautifully  projected  on  the  sky.  Or  the  ring 
systems  may  be  projected  on  a  screen  by  a  polariscope  furnished 
with  concentrating  lenses.  Fig.  352  presents  the  system  as  seen 
through  Iceland  spar  when  the  planes  of  reflection  in  the  polari- 

FIG.  352.  FIG.  353. 


scope  are  at  right  angles.  Two  dark  diameters  cross  the  system 
and  interrupt  the  rings.  If  the  planes  of  reflection  are  coincident, 
the  system  is  in  every  respect  complementary  to  the  other  (Fig. 
353).  The  colors  of  the  rings  are  all  reversed,  and  the  crossing 
bands  are  white.  If  double-refractors  of  two  axes  are  used  instead 
of  the  spar,  compound  systems  are  shown,  of  various  forms  and 
great  beauty. 


CHAPTER    VIII. 

NATURE    OF    LIGHT. —  WAVE    THEORY. 

675.  The  Wave  Theory. — Light  has  sometimes  been  re- 
garded as  consisting  of  material  particles  emanating  from  luminous 
bodies.  But  this,  called  the  corpuscular  or  emission  theory,  has 
mostly  yielded  to  the  undulatory  or  wave  theory,  which  supposes 


400  LIGHT. 

light  to  consist  of  vibrations  in  a  medium.  •  This  medium,  called 
the  luminiferous  ether,  is  imagined  to  exist  throughout  all  space, 
and  to  be  of  such  rarity  as  to  pervade  all  other  matter.  It  is  sup- 
posed also  to  be  elastic  in  a  very  high  degree,  so  that  undulations 
excited  in  it  are  transmitted  with  great  velocity.  If  radiant  heat 
consists  of  undulations  of  the  same  ether,  they  perhaps  differ  from 
those  of  light  only  in  being  slower.  For  it  is  a  familiar  fact,  that 
when  the  heat  of  a  body  is  increased,  a  point  is  at  length  reached 
at  which  the  body  becomes  luminous ;  that  is,  the  vibrations  then 
affect  the  sense  of  sight  as  well  as  that  of  feeling.  Moreover,  the 
rays  of  heat  are  somewhat  less  refrangible  than  those  of  light  (Art. 
625),  from  which  it  is  inferred  that  its  vibrations  are  slower. 

676.  Postulates  of  the  Wave  Theory.— 

1.  The  waves  are  propagated  through  the  ether  at  the  rate  of. 
192,500  miles  per  second. 

As  this  is  the  known  velocity  of  light,  it  must  be  the  rate  aii 
which  the  waves  are  transmitted. 

2.  The  atoms  of  the  ether  vibrate  at  right  angles  to  the  line  of 
the  ray  in  all  possible  directions. 

It  was  at  first  assumed  that  the  luminous  vibrations,  like  the 
vibrations  of  sound,  are  longitudinal,  that  is,  back  and  forth  in  the 
line  of  the  ray ;  but  the  discoveries  in  polarization  require  that  the 
vibrations  of  light  should  be  assumed  to  be  transverse,  that  is,  in 
a  plane  perpendicular  to  the  line  of  the  ray ;  and,  moreover,  that 
in  that  plane  the  vibrations  are  in  every  possible  direction  within 
an  inconceivably  short  space  of  time.  Thus,  if  a  person  is  looking 
at  a  star  in  the  zenith,  we  must  consider  each  atom  of  the  ether 
between  the  star  and  his  eye  as  vibrating  across  the  vertical  in  all 
horizontal  directions,  north  and  south,  east  and  west,  and  in  innu- 
merable lines  between  these. 

3.  Different  colors  are  caused  by  different  rates  of  vibration. 
Red  is  caused  by  the  slowest  vibrations,  and  violet  by  the 

quickest,  and  other  colors  by  intermediate  rates.  White  light  is  to 
the  eye  what  harmony  is  to  the  ear,  the  resultant  effect  of  several 
rates  of  vibration  combined.  There  are  slower  vibrations  of  the 
ether  than  those  of  red  light,  and  quicker  ones  than  those  of  violet 
light,  but  they  are  not  adapted  to  affect  the  vision.  The  former 
affect  the  sense  of  feeling  as  lieat,  the  latter  produce  chemical 
effects,  and  are  called  actinic  rays. 

4.  The  ether  ivitliin  bodies  is  less  elastic  than  in  free  space. 

This  is  inferred  from  the  fact  that  light  moves  with  less  veloc- 
ity in  passing  through  bodies  than  in  free  space ;  the  greater  the 
refractive  power  of  a  body,  the  slower  does  light  move  within  it. 
And  in  some  bodies  of  crystalline  structure,  it  happens  that  the 


REFRACTION    ON    EACH    THEORY.  401 

velocity  is  different  in  different  directions,  so  that  the  elasticity  of 
the  ether  within  them  must  be  regarded  as  varying  with  the 
direction. 

677.  Reflection  and  Refraction  on  the  Wave  Theory. — 

When  the  waves  of  light  reach  the  surface  of  a  new  medium,  the 
ether  within  it  being  generally  in  a  different  state  of  elasticity,  a 
system  of  waves  will  be  propagated  backward  in  the  former  me- 
dium, and  another  onward  in  the  new  medium.  The  reflected 
system  will  make  the  same  angle  with  the  perpendicular  as  the 
incident  system,  analogous  to  the  reflection  of  waves  of  water  and 
of  sound.  But  the  system  which  enters  the  medium  will  change 
its  direction  according  to  its  velocity  in  the  medium ;  and  the 
velocity  depends  on  the  elasticity  of  the  ether.  Pn  media  of  greater 
refractive  power,  the  elasticity  is  considered  to  be  less  than  in 
those  of  less  refractive  power ;  and  the  waves  are  therefore  propa- 
gated more  slowly  in  the  former  than  in  the  latter.  Let  A  B,  d  D 
(Fig.  354),  represent  the  parallel  waves  of  a  beam  falling  on  M  N, 
the  surface  of  a  denser  medium.  The 
side  of  the  wave  which  enters  first  at 
D,  advances  more  slowly  than  the  side  G 

still  moving  in  the  rarer  medium. 
Suppose  D  to  reach  c,  while  d  is  going 
to  0;  then  the  wave,  now  wholly  ==. 
within  the  medium,  lies  in  the  posi-  — 
tion  C  c,  and  advances  in  a  line  per- 
pendicular to  Cc,  so  long  as  it  contin- 
ues in  the  medium.  Thus  the  light 
is  refracted  toward  the  perpendicular 
G  ff,  in  entering  a  denser  medium.  In  a  similar  manner,  it  is 
shown  that  E  F  C  c,  in  entering  a  rarer  medium,  is  refracted  from 
the  perpendicular,  since  the  side  C  d  emerges  first  and  then  gains 
velocity  over  the  side  c  D. 

678.  Refraction  on  the  Emission  Theory. — It  appears, 
therefore,  that  the  wave  theory  requires  us  to  suppose  light  to 
move  more  slowly  in  denser  media.    But,  in  the  emission  the- 
ory, it  is  necessary  to  suppose  it  to  move  more  swiftly.    For  the 
bending  of  the  path  of  a  particle  of  light  toward  the  perpen- 
dicular must  be   attributed  to    the  attraction   exerted  by  the 
medium  on  the  particle.    Suppose,  then,  that  a  particle  of  light 
moves  along  B  A  (Fig.  355),  and  enters  a  denser  medium.     Let 
the  velocity,  B  A,  be  resolved  into  BE,  E A\  the  latter  will  be 
increased  by  the  attraction  of  the  medium ;  the  former  will  not  be 
changed.    Make  A  G  =  B  E  or  D  A,  and  A  F  greater  than  A  E; 
then  A  C  represents  the  direction  and  velocity  of  the  ray  after 

26 


402  LIGHT. 

entering  the  medium.  But  as  A  F  is  greater  than  EA,  while 
A  G  =  D  A,  .*.  A  C  is  greater  than  B  A.  On  the  other  hand,  if  a 
ray,  C  A,  is  entering  a  rarer  medium,  the  F 

attraction  of  the  denser  draws  it  backward, 
and  renders  the  component  A  E  less  than 
F A'  and  hence  the  velocity  A  B,  in  the 
rarer  medium,  is  less  than  C  A,  the  velocity 
in  the  denser.  The  two  theories  are  thus  in 
conflict  on  the  question  whether  light  gains 
or  loses  velocity  in  entering  a  more  refrac- 
tive medium.  Several  direct  tests  have  been 
applied  in  order  to  determine  this ;  and  they 
all  agree  in  proving  that  light  moves  more 
slowly  in  substances  which  have  greater  refractive  power. 

679.  Interference. — Many  interesting  phenomena  are  ex- 
plained on  the  principle  of  interference  of  waves.    As  two  systems 
of  water-waves  may  increase  or  diminish  their  height  by  being 
combined,  and  as  sounds,  when  blended,  may  produce  various 
results,  and  even  destroy  each  other,  so  may  two  pencils  of  light 
either  augment  or  diminish  each  other's  brightness,  and  even  pro- 
duce darkness. 

Any  one  may  try  for  himself  the  following  experiment :  Prick 
two  very  small  holes,  quite  near  each  other,  through  paper,  and 
holding  the  paper  close  to  one  eye,  look  through  both  holes  at  any 
small  bright  spot,  such  as  occurs  in  a  crack  of  glass  when  the  sun 
shines  upon  it ;  then  will  the  bright  spot  be  seen  striped  across 
with  parallel  black  lines,  which  will  be  further  apart  as  the  holes 
are  closer  together.  The  two  pencils  of  light,  through  the  two 
apertures,  overlap  on  the  retina  of  the  eye,  and  cause  bright  and 
dark  lines  by  interference.  Where  like  phases  meet,  the  lines  are 
bright ;  where  opposite  phases  meet,  there  is  no  light,  and  the  lines 
are  black. 

But  there  are  other  forms  of  experiment  by  which  the  exact 
length  of  wave  for  each  color  may  be  determined. 

680.  Interference  by  Thin  Plates. — Let  light  of  any  one 
color,  as  yellow,  fall  on  the  lenses  which  exhibit  Newton's  rings. 
A  system  of  waves  is  reflected  from  the  first  surface  of  the  thin 
stratum  of  air  which  lies  between  the  lenses,  and  another  system 
from  the  second,  and  these  two  come  to  the  eye  together.    Sup- 
pose, at  a  given  point,  the  thickness  of  air  is  such  that  the  reflected 
waves  of  the  second  system  meet  those  of  the  first,  phase  for  phase, 
in  exact  concert ;  at  that  point  is  seen  a  brighter  yellow  than  if 
there  was  but  one  reflecting  surface.    But,  at  another  point,  the 


INTERFERENCE  BY  TWO  MIRRORS.     403 

thickness  of  the  air  may  be  such  that  the  two  systems  disagree  by 
half  a  wave,  bringing  opposite  phases  together ;  in  which  case  all 
motion  is  destroyed,  and  the  point  is  black.  The  former  is  one 
point  of  a  yellow  circle,  the  latter  of  a  black  circle,  each  aror^  " 
the  point  of  contact.  It  is  obvious  that  at  the  smallest  1 
ring,  the  reflected  waves  from  the  second  surface  must  be  just  one 
wave-length  behind  those  from  the  first ;  at  the  second  ring,  two 
waves  behind,  &c. ;  and,  in  general,  luminous  circles  appear  where 
the  two  systems  differ  by  an  exact  number  of  whole  waves,  and 
dark  circles  where  they  differ  by  half  a  wave,  or  any  whole  number 
and  a  half.  The  exact  measurement  of  the  thicknesses  of  air  at 
any  point  (Art.  661),  has  led  to  the  determination  of  the  length  of 
waves  of  each  color. 

681.  Change  of  Color  alters  the  Size  of  the  Rings.— 

If  orange  or  red  light  is  used  instead  of  yellow,  the  rings  are  a  lit- 
tle enlarged,  being  formed  where  the  lamina  of  air  is  a  little 
thicker,  and  therefore  the  waves  for  those  colors  are  longer ;  but 
if  green,  blue,  indigo,  and  violet  are  each  tried  separately,- the 
rings  grow  smaller  in  each  case ;  and  it  is  inferred  that  the  lengths 
of  waves  are  less  in  the  same  order,  and  in  the  ratio  of  the  thick- 
nesses. 

The  reason  becomes  obvious  why,  in  white  light,  the  rings  are 
few  in  number,  and  consist  of  a  series  of  different  colors,  without 
any  black  circles  between.  As  rings  of  different  colors  are  of  dif- 
ferent sizes  when  separate,  so  when  all  colors  are  used  together 
they  will  be  arranged  side  by  side,  and  some  will  be  likely  to  fall 
where  the  black  circles  between  others  would  occur.  Again,  as  all 
the  rings  grow  narrower  at  greater  distances,  because  the  thickness 
of  the  lamina  increases  faster,  they  crowd  upon  and  overlap  each 
other,  and  produce  white  light.  Hence,  a  full  prismatic  series 
occurs  only  near  the  centre,  and  after  five  or  ten  repetitions,  grow- 
ing less  and  less  perfect,  white  light  covers  the  whole  surface. 

682.  Interference  by  Two  Mirrors.— If  two  plane  reflec- 
tors, inclined  at  a  very  obtuse  angle,  receive  light  from  a  minute 
radiant,  and  reflect  it  to  one  spot  on  a  screen,  the  reflected  pencils 
will  interfere,  and  produce  bright  and  dark  lines.     Suppose  light 
of  one  color,  as  violet,  flows  from  a  radiant  point  A  (Fig.  356) ;  let 
mirrors  B  C  and  B  D  reflect  it  to  the  screen  K  L.    F  and  E  may 
be  so  selected  that  the  ray  A  F  +  F  G  equals  the  ray  A  E  +  E  G. 
Then  G  will  be  luminous,  because  the  two  paths  being  equal,  the 
same  phase  of  wave  in  each  ray  will  occur  at  the  point  G.    But  if 
H  be  so  situated  that  Af+fH  differs  half  a  violet  wave  from 
A  e  +  e  H,  then  H  will  be  a  dark  point,  because  opposite  phases 


404 


LIGHT. 


meet  there.  A  similar  point,  7,  will  lie  on  the  other  side  of  6r. 
Again,  there  are  two  points,  K  and  L,  one  on  each  side  of  G,  to 
each  of  which  the  whole  path  of  light  by  one  mirror  will  exceed 

FIG.  356. 


the  whole  by  the  other  by  just  one  violet  wave ;  those  points  are 
bright.  Thus,  there  is  a  series  of  bright  and  dark  points  on  the 
screen ;  or  rather  a  series  of  bright  and  dark  hyperbolic  lines,  of 
which  these  points  are  sections.  Other  colors  will  give  bands  sep- 
arated a  little  further,  indicating  longer  waves.  And  white  light, 
producing  all  these  results  at  once,  will  give  a  repetition  of  the 
prismatic  series. 

683.  Interference  by  Inflection. — One  of  the  forms  of 
inflection  is  explained  as  follows :  Through  an  opaque  screen,  A  B 
(Fig.  357),  let  there  be  a  very  narrow  aperture,  c  d,  by  which  is 
admitted  the  beam  of  light,  efgh,  of  some 
one  color,  and  emanating  from  a  single  point. 
That  part  of  the  aperture  near  d  may  be  re- 
garded as  a  luminous  centre,  from  which  em- 
anate* waves  in  all  directions,  and  the  same  is 
true  of  the  other  part  of  the  aperture  near  c. 
Let  i  be  a  point  on  one  side  of  the  beam,  so 
situated  that  the  distances  d  i  and  c  i  shall 
differ  by  half  a  wave  of  the  color  employed ; 
then,  as  opposite  phases  meet  there,  i  will  be 
a  dark  point.  Let  j  be  a  point  still  further 
removed  from  the  beam,  where  c  j  —  d  j 
equals  the  length  of  a  wave,  then  j  will  be 
luminous,  since  like  phases  meet  in  that 
point.  This  alternation  will  be  repeated  a  few  times  till  the  lumi- 
nous points  become  crowded  and  feeble.  If  the  aperture  is  made 
narrower,  the  intervals  li  i,  i  /,  &c.,  will  increase,  as  they  obviously 
must,  in  order  to  preserve  c  i  —  d  i  equal  to  a  half  wave,  and  cj 
—  dj  equal  to  a  wave.  Violet  light  produces  the  narrowest  lines, 
red  the  widest,  and  white  light  the  prismatic  series,  for  the  same 
reason  as  in  Newton's  rings.  If  A  c,  the  left  side  of  the  screen,  is 
entirely  removed,  so  that  light  passes  only  one  edge,  d,  the  fringes 
will  still  exist,  though  somewhat  modified. 


VIBRATIONS    IN    POLARIZED    LIGHT. 


405 


684.  Length  and  Number  of  Luminous  Waves. — The 

other  cases  of  inflection,  and  the  phenomena  of  striation,  as  well 
as  the  supernumerary  rainbows,  are  fully  accounted  for  on  the 
principle  of  interference.  The  careful  measurements  which  have 
been  made  in  nearly  all  these  instances,  have  led,  by  so  many  in- 
dependent methods,  to  thfe  accurate  determination  of  the  length 
of  a  wave  of  each  color.  When  the  length  of  w,ave  of  any  color 
is  known,  the  number  of  vibrations  per  second  is  readily  obtained 
by  dividing  the  velocity  of  light  by  the  length  of  the  wave,  ffhe 
remarkable  results  of  these  investigations  are  given  in  the  follow- 
ing table : 


Colors. 

Length  of  a  wave 
in  decimals  of 
an  inch. 

Number  of  vibrations  per 
second. 

Extreme  red,       .    . 
Red,      

.OOOO266 
.0000256 

458,OOO,OOO,OOO,OOO 

47  7,000,000,000,000 

Orange,     .... 
Yellow,      .... 
Green,  

.0000240 
.OOO0227 
.OOOO2  I  I 

5o6,OOO,000,000,OOO 
535,000,000,000,000 
577,OOO,OOO,OOO,OOO 

Blue,     

.0000196 

622^OOO,OOO,OOO,OOO 

Indigo,      .... 
Violet,  

.0000185 
.OOOOI74. 

658,000,000,000,000 
699,000,000,000,000 

Extreme  violet,  .     . 

.0000167 

727,000,000,OOO,OOO 

Mean,      .... 

.  000022  s 

54I,OOO,OOO,OOO,OOO 

685.  Change  of  Vibrations  in  Polarized  Light. — It  has 

been  stated  (Art.  676)  that  the  vibrations  of  the  ether,  in  the 
case  of  common  light,  must  be  supposed  to  be  transverse  in  all 
directions.  But,  instead  of  this,  we  may  conceive,  what  is  me- 
chanically equivalent  to  it,  that  the  vibrations  are  made  in  two 
transverse  directions  at  right  angles  to  each  other.  Thus,  in  the 
descent  of  light  from  a  star  in  the  zenith,  we  may  suppose  each 
atom  of  the  ether  to  vibrate  in  the  two  transverse  lines,  one  north 
and  south,  and  the  other  east  and  west ;  because  every  motion 
oblique  to  these  can  be  resolved  into  two  components,  one  on 
each  of  these  two.  Or  any  other  two  lines,  perpendicular  to  each 
other  in  the  same  plane,  may  be  assumed  as  the  directions  of 
vibration. 

This  being  the  nature  of  common  light,  it  is  easy  to  state 
what  is  meant  by  polarized  light.  It  is  that  in  which  the  vibra- 
tions are  performed  in  only  on&  of  the  transverse  directions.  For 
example,  in  the  ray  of  star-light  just  supposed,  if  all  the  easterly 
and  westerly  vibrations,  and  all  the  easterly  and  westerly  compo- 
nents of  the  oblique  vibrations,  were  destroyed,  then  no  motions 


LIGHT. 

would  remain  except  in  the  north  and  south  direction,  and  the 
light  of  that  star  would  be  polarized.  It  is,  of  course,  immaterial 
what  particular  transverse  motion  is  cut  off,  provided  all  the  mo- 
tion at  right  angles  to  it  is  retained. 

686.  Polarizing  and  Analyzing  by  Reflection. — When 
light  is  reflected,  those  vibrations  of  the  ray  which  are  in  the 
plane  of  incidence  are  generally  weakened  in  a  greater  or  less  de- 
gree, while  those  which  are  perpendicular  to  the  same  plane  are 
not  affected.     How  much  the  vibrations  are  weakened  depends  on 
the  elasticity  of  the  ether  within  the  medium,  and  on  the  angle 
of  incidence.     But  reflection  of  light  rarely  if  ever  takes  place 
without  diminishing  the  amplitude  of  those  vibrations  which  are 
in  the  plane  of  incidence ;   so  that  a  reflected  ray  is  always  polar- 
ized, at  least,  in  a  slight  degree. 

It  will  now  be  readily  understood  how  the  analyzing  plate 
(Fig.  350)  proves  the  light  to  be  polarized.  Suppose  the  reflectors 
A  and  C  are  so  perfect  polarizers  that  vibration  in  the  plane  of 
incidence  is  entirely  destroyed.  Along  R  A  the  particles  of  ether 
vibrate  across  it  both  horizontally  and  vertically ;  and  as  the  plane 
of  incidence  R  A  C  is  horizontal,  the  atoms  along  A  C  will 
vibrate  only  vertically,  because  the  horizontal  vibrations,  being 
in  the  plane  of  incidence,  are  destroyed.  Now  let  C  be  placed  so 
as  to  reflect  horizontally ;  the  light  will  not  be  weakened  by  this 
reflection,  because  there  are  no  horizontal  vibrations  to  be  de- 
stroyed. But  let  C  be  turned  so  as  to  reflect  vertically,  for  in- 
stance, upward;  now  there  can  be  no  reflection,  since  all  the 
vibrations  left  in  A  C  are  in  the  vertical  plane,  which  is  the  plane 
of  incidence ;  and  they  are  destroyed.  For  the  same  reason  that 
reflection  at  A  extinguished  all  horizontal  motions  in  the  atoms 
of  ether,  the  reflection  at  C  extinguishes  all  vertical  motions; 
hence  there  is  no  motion  beyond  C. 

687.  Polarizing  by  Transmission  through  a  Bundle  of 
Plates. — At  each  of  the  surfaces  some  reflection  occurs,  so  that 
all  vibrations  in  the  plane  of  incidence  at  length  disappear  from 
the  reflected  ray,  even  though  the  laminae  are  not  perfect  polar- 
izers ;  while  all  vibrations  perpendicular  to  this  plane  are  pre- 
served.   Hence  the  reverse  must  be  true  of  the  transmitted  ray ; 
it  will  retain  the  vibrations,  so  far  as  they  coincide  with  the  plane 
of  incidence,  and  lose  them,  so  far  as  they  are  perpendicular  to  it. 
Thus  the  two  sets  of  rectangular  vibrations  are  separated  from 
each  other ;  one  exists  in  the  reflected  ray,  the  other  in  the  trans- 
mitted ray.    The  two  rays  are  therefore  polarized  in  planes  at 
right  angles  to  each  other. 


IMAGE    BY    AN    APERTURE.  .407 

688.  Polarizing  by  Absorption.— A  tourmaline  absorbs,  or 
in  some  way  extinguishes  the  vibrations,  so  far  as  they  are  perpen- 
dicular to  its  crystalline  axis,  but  leaves  all  motion  which  is  par- 
allel to  its  axis  unimpaired.     It  is  at  once  apparent  why  a  second 
tourmaline  analyzes ;  for  if  its  axis  is  parallel  to  that  of  the  first, 
the  same  vibrations  which  could  pass  the  one,  could  pass  the  other 
also ;  but  if  the  two  axes  are  at  right  angles,  the  same  system  of 
vibrations  which  could  pass  the  first,  because  parallel  to  its  axis, 
will  be  absorbed  by  the  second,  because  perpendicular  to  its  axis. 

689.  Polarizing  by  Double  Refraction. — In  doubly-re- 
fracting crystals,  the  ether  possesses  different  degrees  of  elasticity 
in  different  directions ;  hence,  so  far  as  vibrations  lie  in  one  plane, 
they  may  be  more  retarded  in  their  progress,  and  in  a  plane  at 
right  angles  to  that  they  may  be  less  retarded,  and  the  degree  of 
refraction  depends  on  the  amount    of  retardation   (Art   677). 
Thus  the  two  systems  become  separated,  and  emerge  at  different 
places.    Each  ray  is  of  course  polarized,  having  vibrations  in  only 
one  direction ;  and  the  two  planes  of  polarization  are  at  right 
angles  to  each  other. 

690.  Different  Kinds  of  Polarization. — Since  the  discovery 
was  made  that  the  etherial  atoms  may  by  certain  methods  be 
thrown  into  circular  movements,  and  by  others  into  vibrations  in 
an  ellipse  with  the  axis  in  a  fixed  direction,  the  polarization 
already  described  has  been  called  plane  polarization,  since  the 
atoms  vibrate  in  a  plane.     Circular  polarization  is  that  in  which 
the  atoms  revolve  in  circles ;  and  elliptical  polarization  denotes  a 
state  of  vibration  in  ellipses,  whose  major  axes  are  confined  to  one 
plane. 


CHAPTER    IX. 

VISION. 

691.  Image  by  Light  through  an  Aperture.— If  light  from 
an  external  object  pass  through  a  small  opening  of  any  shape  in 
the  wall  of  a  dark  room,  it  will  form  an  ill-defined  inverted  image 
on  the  opposite  wall.  Imagine  a  minute  square  orifice,  through 
which  the  light  enters  and  falls  on  a  screen  several  feet  distant. 
A  pencil  of  light,  in  the  shape  of  a*  square  pyramid,  emanating  from 
the  highest  point  of  the  object,  passes  through  the  aperture,  and 


408  LIGHT. 

forms  a  luminous  square  near  the  bottom  of  the  screen.  From  an 
adjacent  point  another  pencil,  crossing  the  first  at  the  aperture, 
forms  another  square,  overlapping  and  nearly  coinciding  with  the 
former.  Thus  every  point  of  the  object  is  represented  by  its 
square  on  the  screen ;  and  as  the  pencils  all  cross  at  the  aperture, 
the  image  formed  is  every  way  inverted.  It  is  also  indistinct, 
because  the  squares  overlap,  and  the  light  of  contiguous  points  is 
mingled  together.  If  the  orifice  is  smaller,  the  image  is  less  lumi- 
nous, but  more  distinct,  because  the  pencils  which  form  it  overlap 
in  a  less  degree.  If  the  hole  is  circular,  or  triangular,  or  of  irreg- 
ular form,  there  is  no  change  in  the  appearance  of  the  image, 
which  is  now  composed  of  small  circles,  or  triangles,  or  irregular 
figures,  whose  shape  is  completely  lost  by  overlapping. 

692.  Effect  of  a  Convex  Lens  at  the  Aperture.— The 

image  will  become  distinct,  and  more  luminous  also,  if  the  aper- 
ture be  enlarged  to  a  diameter  of  two  or  three  inches,  and  then 
covered  by  a  convex  lens  of  the  proper  curvature.  The  image  will 
be  distinct,  because  the  rays  from  each  point  of  the  object  are  con- 
verged to  a  point  again,  and  luminous,  in  proportion  as  the  lens 
has  a  larger  area  than  the  aperture  before  employed.  This  is  a 
real,  and  therefore  an  inverted  image  (Art.  618).  A  scioptic  ball 
is  a  sphere  containing  a  lens,  and  so  fitted  in  a  socket  that  it  can 
be  turned  in  any  direction,  and  thus  bring  into  the  room  the  im- 
ages of  different  parts  of  the  landscape.  The  camera  obscura  is  a 
darkened  room  furnished  with  a  scioptic  ball  and  adjustable  screen 
for  producing  distinct  pictures  of  external  objects. 

Instead  of  connecting  the  lens  with  the  wall  of  a  room,  it  is 
frequently  attached  to  a  portable  box  or  case,  within  which  the 
image  is  formed.  The  Daguerreotype,  or  photograph,  is  the  image 
produced  by  the  convex  lens,  and  rendered  permanent  by  the 
chemical  action  of  light  on  a  surface  properly  prepared.  The  lens 
for  photographic  purposes  needs  to  be  achromatic,  and  corrected, 
also,  as  far  as  possible,  for  spherical  aberration. 

693.  The  Bye. — The  eye  is  a  camera  obscura  in  miniature; 
we  find  here  the  darkened  room,  the  aperture,  the  convex  lens,  and 
the  screen,  with  inverted  images  of  external  objects  painted  on  it. 
A  horizontal  section  of  the  eye  is  represented  in  Fig.  358. 

The  optical  apparatus  of  the  eye,  and  the  spherical  case  which 
incloses  it,  constitute  what  is  called  the  eye-ball.  The  case  itself, 
except  about  a  sixth  part  of  it  in  front,  is  a  strong  white  substance, 
called,  on  account  of  its  hardness,  the  sclerotic  coat,  8,  $(Fig.  358). 
In  the  front,  this  opaque  coat  changes  to  a  perfectly  transparent 
covering,  called  the  cornea,  C,  C,  .which  is  a  little  more  convex  than 
the  sclerotic  coat.  The  increased  convexity  of  the  "cornea  may  be 


THE    INTERIOR    OF    THE    EYE 


409 


felt  by  laying  the  finger  gently  on  the  eye-lid  when  closed,  and 

then    rolling    the  eye 

one  way  and  the  other.  FlG-  35& 

The  bony  socket, 
which  contains '  the 
eye,  is  of  pyramidal 
form,  its  vertex  being 
some  distance  behind 
the.  eye-ball;  room  is 
thus  afforded  for  the 
mechanism  which  gives 
it  motion.  This  cav- 
ity, except  the  hemi- 
sphere in  front  occu- 
pied by  the  eye  itself, 
is  filled  up  with  fatty 
matter  and  with  the 
six  muscles  by  which 
the  eye-ball  is  revolved  in  all  directions. 

694.  The  Interior  of  the  Eye. — Behind  the  cornea  is  a 
fluid,  A,  called  the  aqueous  humor.  In  the  back  part  of  this  fluid 
lies  the  iris,  I,  /,  an  opaque  membrane,  having  in  the  centre  of  it 
a  circular  aperture,  the  pupil,  through  which  the  light  enters. 
The  iris  is  the  colored  part  of  the  eye ;  the  back  side  of  it  is  black. 
Directly  back  of  the  aqueous  humor  and  iris,  is  a  flexible  double 
convex  lens,  L,  called  the  crystalline  lens,  or  crystalline  humor, 
having  the  greatest  convexity  on  the  back  side.  The  large  space 
back  of  the  crystalline  is  occupied  by  the  vitreous  humor,  V,  a 
semi-liquid,  of  jelly-like  consistency,  Next  to  the  vitreous  humor 
succeed  those  inner  coatings  of  the  eye,  which  are  most  immedi- 
ately concerned  in  vision.  First  in  order  is  the  retina,  R,  R,  on 
which  the  light  paints  the  inverted  pictures  of  external  objects. 
The  fibres  of  the  optic  nerve,  which  enter  the  ball  at  _ZV^  are  spread 
all  over  the  retina,  and  convey  the  impressions  produced  there  to 
the  brain.  Outside  of  the  retina  is  the  choroid  coat,  c  h,  c  Ji,  cov- 
ered with  a  black  pigment,  which  serves  to  absorb  all  the  light  so 
soon  as  it  has  passed  through  the  retina  and  left  its  impressions. 
The  choroid  is  inclosed  by  the  sclerotic  already  described.  The 
nerve-fibres,  which  are  spread  over  the  interior  of  the  retina,  are 
gathered  into  a  compact  bundle  about  one-tenth  of  an  inch  in 
diameter,  which  passes  out  through  the  three  coatings  at  the  back 
part  of  the  ball,  about  fifteen  degrees  from  the  axis,  X  X,  on  the 
side  toward  the  other  eye.  M,  M  represent  two  of  the  muscles, 
where  they  are  attached  to  the  eye-ball. 


410  LIGHT. 

695.  Vision. — The  index  of  refraction  for  the  cornea,  and  the 
aqueous  and  vitreous  humors,  is  just  about  the  same  as  that  for 
water;  for  the  crystalline  lens,  the  index  is  a  little  greater.    The 
light,  therefore,  which  conies  from  without,  is  converged  principally 
on  entering  the  cornea,  and  this  convergency  is  a  little  increased 
both  on  entering  and  leaving  the  crystalline.    If  the  convergency 
is  just  sufficient  to  bring  the  rays  of  each  pencil  to  a  focus  on  the 
retina,  then  the  images  are  perfectly  formed,  and  there  is  distinct 
vision.    To  prevent  the  reflection  of  rays  back  and  forth  within 
the  chamber  of  the  eye,  its  walls  are  made  perfectly  black  through- 
out by  a  pigment  which  lines  the  choroid,  the  ciliary  processes, 
and  the  back  of  the  iris.     Telescopes  and  other  optical  instru- 
ments are  painted  black  in  the  interior  for  a  similar  purpose. 

The  cornea  is  prevented  from  producing  spherical  aberration 
by  the  form  of  a  prolate  spheroid  which  is  given  to  its  surface,  and 
the  crystalline,  by  a  gradual  increase  of  density  ffom  its  edge  to 
its  centre. 

696.  Adaptations. — By  the  prominence  of  the  cornea  rays 
of  considerable  obliquity  are  converged  into  the  pupil,  so  that  the 
eye,  without  being  turned,  has  a  range  of  vision  more  or  less  per- 
fect, through  an  angle  of  about  150°. 

The  quantity  of  light  admitted  into  the  eye  is  regulated  by  the 
size  of  the  pupil.  The  iris,  composed  of  a  system  of  circular  and 
radial  muscles,  expands  or  contracts  the  pupil  according  to  the 
intensity  of  the  light.  These  changes  are  involuntary ;  a  person 
may  see  them  in  his  own  eyes  by  shading  them,  and  again  letting 
a  strong  light  fall  upon  them,  while  he  is  before  a  mirror. 

The  pupils  in  the  eyes  of  animals  have  different  forms  accord- 
ing to  their  habits ;  in  the  eyes  of  those  which  graze,  the  pupil  is 
elongated  horizontally,  and  in  the  eyes  of  beasts  and  birds  of 
prey,  it  is  elongated  vertically. 

The  eyes  of  animals  are  adapted,  in  respect  to  their  refractive 
power,  to  the  medium  which  surrounds  them.  Animals  which 
inhabit  the  water  have  eyes  which  refract  much  more  than  those 
of  land  animals.  The  human  eye  being  fitted  for  seeing  in  air,  is 
unfit  for  distinct  vision  in  water,  since  its  refractive  power  is 
nearly  the  same  as  that  of  water,  and  therefore  a  pencil  of  parallel 
rays  from  water  entering  the  eye  would  scarcely  be  converged  at 
all.  The  effect  is  the  same  as  if  the  cornea  were  deprived  of  all 
its  convexity. 

697.  Accommodation  to  Diminished  Distance.— It  has 

been  shown  (Art.  618),  that  as  an  object  approaches  a  lens,  its 
image  moves  away,  and  the  reverse.    Therefore  in  the  eye  there 


LONG  SIGHTEDNESS.  411 

must  be  some  change  in  order  to  prevent  this,  and  keep  the  image 
distinct  on  the  retina  while  the  object  varies  its  distance.  In  a 
state  of  rest,  the  eye  converges  to  the  retina  only  the  pencils  of 
parallel  rays,  that  is,  those  which  come  from  objects  at  great  dis- 
tances. Rays  from  near  objects  diverge  so  much  that,  while  the 
eye  is  at  rest,  it  cannot  sufficiently  converge  them  so  that  they 
will  meet  on  the  retina ;  but  each  conical  pencil  is  cut  off  before 
reaching  its  focus,  and  all  the  points  of  the  object  are  represented 
by  overlapping  circles,  causing  an  indistinct  image.  The  change 
in  the  eye,  which  fits  it  for  seeing  near  objects  distinctly,  is  called 
accommodation.  This  is  effected  by  increasing  the  convexity  of 
the  crystalline  lens,  principally  the  front  surface,  The  ciliary 
muscle,  m,  m,  surrounds  the  crystalline,  and  is  attached  to  the 
sclerotic  coat  just  on  the  circle  where  it  changes  into  the  cornea. 
This  muscle  is  connected  with  the  edge  of  the  crystalline  by  the 
circular  ligament  which  surrounds  the  latter  and  holds  it  in  place. 
When  the  muscle  contracts,  it  relaxes  the  ligament,  and  the  crys- 
talline, by  its  own  elastic  force,  begins  to  assume  a  more  convex 
form,  as  represented  by  the  dotted  line.  The  eye  is  then  accom- 
modated for  the  vision  of  objects  more  or  less  near, 
according  to  the  degree  of  change  in  the  lens.  On  FlG- 
the  other  hand,  when  the  ciliary  muscle  relaxes,  the 
ligament  again  draws  upon  the  lens  to  flatten  it,  and 
adapt  it  for  the  view  of  distant  objects.  In  Fig.  359 
these  two  conditions  of  the  crystalline  are  more  dis- 
tinctly shown.  The  dotted  line  exhibits  the  shape 
of  the  lens  when  accommodated  for  seeing  near  ob- 
jects. Accompanying  this  action  of  the  ciliary  mus- 
cle is  that  of  the  iris,  which  diminishes  the  pupil  for 
near  objects,  so  as  to  exclude  the  outer  and  more  divergent  rays. 
The  dotted  lines  in  front  of  the  iris  represent  its  situation  when 
pushed  forward  by  the  crystalline  accommodated  for  near  objects. 

698.  Long-Sightedness. — As  life  advances,  the  crystalline 
becomes  harder  and  less  elastic.  It  therefore  assumes  a  less  con- 
vex form  when  the  ligament  is  relaxed,  and  cannot  be  accommo- 
dated to  so  short  distances  as  in  earlier  years ;  and  at  length  it 
remains  so  flattened  in  shape  that  only  very  distant  objects  can 
be  seen  distinctly.  The  eye  is  then  said  to  be  long-sighted,  and 
requires  a  convex  lens  to  be  placed  before  it,  to  compensate  for 
insufficient  convexity  in  the  crystalline. 

There  are,  however,  cases  of  long-sightedness  in  early  life. 
Such  instances  are  found  to  be  the  result  of  an  oblate  form  of  the 
eye-ball,  as  shown  in  Fig.  360 ;  it  is  too  short  from  front  to  back 
to  furnish  room  for  the  convergency  of  the  pencils,  and  they  are 


412  LIGHT. 

cut  off  by  the  retina  before  reaching  their  focal  points.  In  order 
to  bring  the  distinct  image  forward  upon  the  retina,  convex 
glasses  are  needed  in  such 

cases,  just  as  for  the  eyes  FIG.  360. 

of  most  people  when  ad- 
vanced in  life.  As  the  term 
long-sightedness  is  now  ap- 
plied to  this  abnormal  con- 
dition of  the  eye,  the  effect  of 
age  upon  the  sight  is  more 
properly  called  old-sightedness. 

699.  Short  -  Sightedness.— The  eyes  of  the  short-sighted 
have  a  form  the  reverse  of  that  just  described ;  the  eye-ball  is 
elongated  from  cornea  to  retina  (Fig.  361),  resembling  a  prolate 
spheroid,  so  that  rays  parallel,  or  nearly  so,  are  converged  to  a 
point  before  reaching  the  retina,  and  after  crossing,  fall  on  it  in  a 
circle ;  and  the  image,  made 

up  of  overlapping  circles  in-  FlG- 

stead  of  points,  is  indistinct. 
If  this  elongation  of  the  eye- 
ball is  extreme,  an  object 
must  be  brought  very  near, 
in  order  that  its  image  may 
move  back  to  the  retina,  and 
distinct  vision  be  produced. 

This  inconvenience  is  remedied  by  the  use  of  concave  lenses, 
which  increase  the  divergency  of  the  rays  before  they  enter  the 
eye,  and  thus  throw  their  focal  points  further  back. 

In  the  normal  condition  of  the  eyes  in  early  life,  the  nearest 
limit  of  distinct  vision  is  about  five  inches.  This  limit  slowly 
increases  with  advance  of  life,  but  much  more  slowly  in  some 
cases  than  others,  till  it  is  at  an  indefinitely  great  distance.  The 
near  limit  of  distinct  vision  for  the  short-sighted  varies  from  five 
down  to  two  inches,  according  to  the  degree  of  elongation  in  the 
eye-ball. 

700.  Why  an  Object  is  Seen  Erect  and  Single.— The 

image  on  the  retina  is  inverted  ;  and  that  is  the  very  reason  why 
the  object  is  seen  erect ;  the  image  is  not  the  thing  seen,  but  that 
ly  means  of  which  we  see.  The  impression  produced  at  any  point 
on  the  retina  is  referred  outward  in  a  straight  line  through  a  point 
near  the  centre  of  the  lens,  to  something  external  as  its  cause ; 
and  therefore  that  is  judged  to  be  highest  without  us  which  makes 
its  image  lowest  on  the  retina,  and  the  reverse. 


CONTINUANCE    OF    IMPRESSIONS.  413 

An  object  appears  as  one,  though  we  see  it  by  means  of  two 
images ;  but  this  is  only  one  of  many  instances  in  which  we  have 
learned  by  experience  to  refer  two  or  more  sensations  to  one  thing 
as  the  cause.  Provided  the  images  fall  on  parts  of  the  retina, 
which  in  our  ordinary  vision  correspond  with  each  other,  then  by 
experience  we  refer  both  impressions  to  one  object ;  but  if  we 
press  one  eye  aside,  the  image  falls  in  a  new  place  in  relation  to 
the  other,  and  the  object  seems  double. 

701.  Indirect  Vision.— The  Blind  Point— To  obtain  a 
clear  and  satisfactory  view  of  an  object,  the  axes  of  both  eyes  are 
turned  directly  upon  it,  in  which  case  each  image  is  at  the  centre 
of  the  retina.    But  when  the  light  from  an  object  is  exceedingly 
faint,  it  is  better  seen  by  indirect  vision,  that  is,  by  looking  to  a 
point  a  little  on  one  side,  and  especially  by  changing  the  direction 
of  the  eyes  from  moment  to  moment,  so  that  the  image  may  fall 
in  various  places  near  the  centre  of  the  retina.    Many  heavenly 
bodies  are  plainly  discerned  by  indirect  vision,  which  are  too  faint 
to  be  seen  by  direct  vision. 

In  the  description  of  the  eye  it  was  stated  that  the  retina,  as 
well  as  the  choroid  and  the  sclerotic,  is  perforated  to  allow  the 
optic  nerve  to  pass  through.  At  that  place  there  is  no  vision,  and 
it  is  called  the  Uind  point.  In  each  eye  ifc  is  situated  about  15° 
from  the  centre  of  the  retina  toward  the  other  eye.  Let  a  person 
close  his  right  eye,  and  with  the  left  look  at  a  small  but  conspicu- 
ous object,  and  then  slowly  turn  the  eye  away  from  it  toward  the 
right ;  presently  the  object  will  entirely  disappear,  and  as  he  looks 
still  further  to  the  right,  it  will  after  a  moment  reappear,  and  con- 
tinue in  sight  till  the  axis  of  the  eye  is  turned  70°  or  80°  from  it. 
The  same  experiment  may  be  tried  with  the  right  eye  in  the  oppo- 
site direction.  The  reason  why  people  do  not  generally  notice  the 
fact  till  it  is  pointed  out,  is  that  an  object  cannot  disappear  to 
both  eyes  at  once,  nor  to  either  eye  alone,  when  directed  to  the 
object. 

702.  Continuance  of  Impressions. — The  impression  which 
a  visible  object  makes  upon  the  retina  continues  about  one-eighth 
or  one-ninth  of  a  second ;  so  that  if  the  object  is  removed  for 
that  length  of  time,  and  then  occupies  its  place  again,  the  vision 
is  uninterrupted.    A  coal  of  fire  whirled  round  a  centre  at  the 
rate  of  eight  or  nine  times  per  second,  appears  in  all  parts  of  the 
circumference  at  once.     When  riding  in  the  cars,  one  sometimes 
gets  a  faint  but  apparently  an  uninterrupted  view  of  the  landscape 
beyond  a  board  fence,  by  means  of  successive  glimpses  seen 
through  the  cracks  between  the  upright  boards.    Two  pictures, 


414  LIGHT. 

on  opposite  sides  of  a  disk,  are  brought  into  view  together,  as 
parts  of  one  and  the  same  picture,  by  whirling  the  disk  rapidly 
on  one  of  its  diameters.  Such  an  instrument  is  called  a  tliauma- 
trope.  The  pJiantasmascope  is  constructed  on  the  same  principle. 
Several  pictures  are  painted  in  the  sectors  of  a  circular  disk,  rep- 
resenting the  same  object  in  a  series  of  positions.  These  are 
viewed  in  a  mirror  through  holes  in  the  disk,  as  it  revolves  quickly 
in  its  own  plane.  Each  glimpse  which  is  caught  whenever  a  hole 
comes  before  the  eye,  presents  the  object  in  a  new  attitude ;  and 
all  these  views  are  in  such  rapid  succession  that  they  appear  like 
one  object  going  through  the  series  of  movements. 

703.  Accidental   Colors. — There  are  impressions  on  the 
retina  of  another  kind,  which  are  produced  by  intense  lights;  they 
continue  longer,  and  are  in  respect  to  color  unlike  the  objects 
which  cause  them.    They  are  commonly  called  accidental  colors. 
If  a  particular  part  of  the  retina  is  for  some  time  affected  by  the 
image  of  a  bright  colored  object,  and  then  the  eyes  are  shut,  or 
turned  upon  a  white  surface,  the  form  appears  to  remain,  but  the 
color  is  complementary  to  that  of  the  object ;  and  its  continuance 
is  for  a  few  seconds  or  several  minutes,  according  to  the  vividness 
of  the  impression.    This  is  the  cause  of  the  green  appearance  of 
the  sky  between  clouds  of  brilliant  red  in  the  morning  or  evening. 

704.  Estimate  of  the  Distance  of  Bodies.— 

1.  If  objects  are  near,  we  judge  of  relative  distance  by  the  in- 
clination of  the  optic  axes  to  each  other.     The  greater  that  incli- 
nation is,  or,  which  is  the  same  thing,  the  greater  the  change  of 
direction  in  an  object,  as  it  is  viewed  by  one  eye  and  then  by  the 
other,  the  nearer  it  is.    If  objects  are  very  near,  we  can  with  one 
eye  alone  judge  of  their  distance  by  the  degree  of  effort  required  to 
accommodate  the  eye  to  that  distance. 

2.  If  objects  are  known,  we  estimate  their  distance  by  the  visual 
angle  which  they  fill,  having  by  experience  learned  to  associate 
together  their  distance  and  their  apparent,  that  is,  their  angular 
size. 

3.  Our  judgment  of  distant  objects  is  influenced  by  their  clear- 
ness or  obscurity.    Mountains,  and  other  features  of  a  landscape, 
if  seen  for  the  first  time  when  the  air  is  remarkably  pure,  are  esti- 
mated by  us  nearer  than  they  really  are ;  and  the  reverse,  if  the 
air  is  unusually  hazy. 

4.  Oar  estimate  of  distance  is  more  correct  when  many  objects 
intervene.     Hence  it  is  that  we  are  able  to  place  that  part  of  the 
sky  which  is  near  the  horizon  further  from  us  than  that  which  is 
over  our  heads.    The  apparent  sky  is  not  a  hemisphere,  but  a  flat- 
tened semi-ellipsoid. 


;  BINOCULAR    VISION.  415 

705.  Magnitude  and  Distance  Associated. — Our  judg- 
ments of  distance  and  of  magnitude  are  closely  associated.    If 
objects  are  known,  we  estimate  their  distance  by  their  visual  angle, 
as  has  been  stated ;  but  if  unknown,  we  must  first  acquire  our 
notion  of  their  distance  by  some  other  means,  and  then  their  visual 
angle  gives  us  a  definite  impression  as  to  their  size.    And  if  our 
judgment  of  distance  is  erroneous,  a  corresponding  error  attaches 
to  our  estimate  of  their  magnitude.    An  insect  crawling  slowly 
on  the  window,  if  by  mistake  it  is  supposed  to  be  some  rods  be- 
yond the  window,  will  appear  like  a  bird  flying  in  the  air.    The 
moon  near  the  horizon  seems  larger  than  above  us,  because  we  are 
able  to  locate  it  at  a  greater  distance. 

706.  Binocular  Vision. — The  Stereoscope. — If  objects  are 
placed  quite  near  us,  we  obtain  simultaneously  two  views,  which 
are  essentially  different  from  each  other — one  with  one  eye,  and 
one  with  the  other.    By  the  right  eye  more  of  the  right  side,  and 
less  of  the  left  side,  is  seen,  than  by  the  left  eye.    Also,  objects  in 
the  foreground  fall  further  to  the  left  compared  with  distant  ob- 
jects, when  seen  with  the  right  eye  than  when  seen  with  the  left. 
And  we  associate  with  th'ese  combined  views  the  form  and  extent 
of  a  body,  or  group  of  bodies,  particularly  in  respect  to  distance  of 
parts  from  us.    It  is,  then,  by  means  of  vision  with  two  eyes,  or 
binocular  vision,  that  we  are  enabled  to  get  accurate  perceptions 
of  prominence  or  depression  of  surface,  reckoned  in  the  visual 
direction.    A  picture  offers  no  such  advantage,  since  all  its  parts 
are  on  one  surface,  at  a  common  distance  from  the  eyes.    But,  if 
two  perspective  views  of  an  object  should  be  prepared,  differing  as 
those  views  do,  which  are  seen  by  the  two  eyes,  and  if  the  right 
eye  could  then  see  only  the  right-hand  view,  and  the  left  eye  only 
the  left-hand  view,  and  if,  furthermore,  these  two  views  could  be 
made  to  appear  on  one  and  the  same  ground,  the  vision  would 
then1  be  the  same  as  is  obtained  of  the  real  object  by  both  eyes. 
This  is  effected  by  the  stereoscope.     Two  photographic  views  are 
taken,  in  directions  which  make  a  small  angle  with  each  other,  and 
these  views  are  seen  at  once  by  the  two  eyes  respectively,  through 
a  pair  of  half-lenses,  placed  with  their  thin  edges  toward  each 
other,  so  as  to  turn  the  visual  pencils  away  from  each  other,  as 
though  they  emanated  from  one  object.    An  appearance  of  relief 
and  reality  is  thus  given  to  superficial  pictures,  precisely  like  that 
obtained  from  viewing  the  objects  themselves. 


416 


LIGHT. 


FIG.  362. 


CHAPTER    X. 

OPTICAL    INSTRUMENTS. 

707.  The  Camera  Lucida. — This  is  a  four-sided  prism,  so 
contrived  as  to  form  an  apparent  image  at  a  surface  on  which  that 
image  may  be  copied,  the  surface  and  image  being  both  visible  at 
the  same  time.  It  has  the  form  represented  by  the  section  in  Fig. 
362 ;  A  =  90°,  C  =  135° ;  B  and  D, 
of  any  convenient  size,  their  sum  of 
course  =  135°.  A  pencil  of  light 
from  the  object  M,  falling  perpendic- 
ularly on  A  D,  proceeds  on,  and 
makes,  with  D  C,  an  angle  equal  to 
the  complement  of  D.  After  suffer- 
ing total  reflection  at  G,  and  again  at 
H,  its  direction  H  E  is  perpendicular 
to  MF.  For,  produce  MF  and  EH, 
till  they  intersect  in  /;  then,  since 
C  =  135°,  C  G  H+  GH  G  =  45° ;  but 
IGH=2CGfi,  and  I H  G  = 
2CHG-,  .-.  IGH+  IHG  =  W°', 
.'.  I  =  90°.  Therefore  HE  emerges  at  right  angles  to  A  B,  and  is 
not  refracted.  Now,  if  the  pupil  of  the  eye  be  brought  over  the 
edge  B,  so  that,  while  E  ^Tenters,  there  may  also  enter  a  pencil 
from  the  surface  at  M'9  then  both  the  surface  M'  and  the  object 
M  will  be  seen  coinciding  with  each  other,  and  the  hand  may 
therefore  sketch  M  on  the  surface  at  M' .  The  reason  for  two 
reflections  of  the  light  is,  that  the  inversion  produced  by  one 
reflection  may  be  restored  by  the  second. 

One  of  the  most  useful  applications  of  the  camera  lucida  is  in 
connection  with  the  compound  microscope,  where  it  is  employed 
in  copying  with  exactness  the  forms  of  natural  objects,  too  small 
to  be  at  all  visible  to  the  naked  eye. 

708.  The 'Microscope. — This  is  an  instrument  for  viewing 
minute  objects.  The  nearer  an  object  is  brought  to  the  eye,  the 
larger  is  the  angle  which  it  fills,  and  therefore  the  more  perfect  is 
the  view,  provided  the  rays  of  each  pencil  are  converged  to  a  point 
on  the  retina.  But  if  the  object  is  nearer  than  the  limit  of  dis- 
tinct vision,  the  eye  is  unable  to  produce  sufficient  convergency. 
If  the  letters  of  a  book  are  brought  close  to  the  eye,  they  become 
blurred  and  wholly  illegible.  But  let  a  pin-hole  be  pricked  through 


THE    COMPOUND    MICROSCOPE.  417 

paper,  and  interposed  between  the  eye  and  the  letters,  and,  though 
faint,  they  are  distinct  and  much  enlarged.  The  distinctness  is 
owing  to  the  fact  that  the  outer  rays,  which  are  most  divergent, 
are  excluded,  and  the  eye  is  able  to  converge  the  few  central  rays 
of  each  pencil  to  a  focus.  The  letters  appear  magnified,  because 
they  are  so  near,  and  fill  a  large  angle.  The  microscope  utilizes 
these  excluded  rays,  and  renders  the  image  not  only  large  and  dis- 
tinct, but  luminous. 

709.  The  Single  Microscope. — The  single  microscope  is 
merely  a  convex  lens.    It  aids  the  eye  in  converging  the  rays, 
which  come  from  a  very  near  object,  so  that  a  distinct  and  lumin- 
ous image  may  be  formed  on  the  retina.     The  lens  may  be  re- 
garded as  a  part  of  the  eye,  and  the  diameter  of  an  object  is  mag- 
nified in  the  ratio  of  the  limit  of  distinct  vision  to  the  focal 
distance  of  the  lens.    Taking  five  inches  as  the  limit  of  distinct 
vision,  if  the  principal  focal  distance  is  one-fourth  of  an  inch,  then 
we  may  consider  the  object  twenty  times  nearer  the  eye  than  in 
viewing  it  without  a  lens,  and  therefore  magnified  twenty  times  in 
diameter,  or  400  times  in  area.    Now  glass  lenses  are  made  whose 
focal  length  is  not  more  than  ^  inch,  and  whose  magnifying 
power,  therefore,  is  5  :  •£•$,  =  250  in  diameter,  or  62,500  in  area. 

Though  the  focal  distance  of  a  lens  may  be  made  as  small  as 
we  please,  yet  a  practical  limit  to  the  magnifying  power  is  very 
soon  reached. 

1.  The  field  of  view,  that  is,  the  extent  of  surface  which  can 
be  seen  at  once,  diminishes  as  the  power  is  increased. 

2.  Spherical  aberration  increases  rapidly,  because  the  outer 
rays  are  very  divergent.    Hence  the  necessity  of  diminishing  the 
aperture  of  the  lens,  in  order  to  exclude  the  most  divergent  rays. 

3.  It  is  more  difficult  to  illuminate  the  object  as  the  focal 
length  of  the  lens  becomes  less;   and  this  difficulty  becomes  a 
greater  evil  on  account  of  the  necessity  of  diminishing  the  aper- 
ture in  order  to  reduce  the  spherical  aberration. 

Magnifying  glasses  are  single  microscopes  of  low  power,  such 
as  are  used  by  watchmakers.  Lenses  of  still  lower  power  and 
several  inches  in  diameter  are  used  for  viewing  pictures. 

710.  The  Compound  Microscope.— It  is  so  called  because 
it  consists  of  two  parts,  an  object-glass,  by  which  a  real  and  mag- 
nified image  is  formed,  and  an  eye-glass,  by  which  that  image  is 
again  magnified.    Its  general  principle  may  be  explained  by  Fig. 
363,  in  which  a~b  is  the  small  object,  cd  the  object-glass,  and  ef 
the  eye-glass.    Let  a  b  be  a  little  beyond  the  principal  focus  of  c  d, 
and  then  the  image  gh  will  be  real,  on  the  opposite  side  of  cd, 

27 


418 


LIGHT. 


and  larger  than  a  1).  Now  apply  ef  as  a  single  microscope  for 
viewing  g  h,  as  though  it  were  an  object  of  com- 
paratively large  size.  Let  g  h  be  at  the  princi- 
pal focus  of  ef,  so  that  the  rays  of  each  pencil 
shall  be  parallel ;  they  will,  therefore,  come  to 
the  eye  at  &,  from  an  apparent  image  on  the 
same  side  as  the  real  one,  g  h ;  and  the  extreme 
pencils,  eJc,fk,  if  produced  backward,  will  in- 
clude the  image  between  them,  e  kf  being  the 
angle  which  it  fills. 

711.  The  Magnifying  Power. — The  mag- 
nifying power  of  the  compound  microscope  is 

estimated  by  compounding  two  ratios ;  first,  the  distance  of  the 
image  from  the  object-glass,  to  the  distance  of  the  object  from  the 
same ;  and  secondly,  the  limit  of  distinct  vision  to  the  distance 
of  the  image  from  the  eye-glass.  For  the  image  itself  is  enlarged 
in  the  first  ratio  (Art.  618) ;  and  the  eye-glass  enlarges  that  image 
in  the  second  ratio  (Art.  709).  The  advantage  of  this  form  over 
the  single  microscope  is  not  so  much  that  a  great  magnifying 
power  is  obtained,  as  that  a  given  magnifying  power  is  accom- 
panied by  a  larger  field  of  view. 

712.  Modern  Improvements. — Great  improvements  have 
been  made  in  the  compound  microscope,  principally  by  combining 
lenses  in  such  a  manner  as  greatly  to  reduce  the  chromatic  and 
spherical  aberrations.    The  object-glass  generally  consists  of  one, 
two,  or  three  achromatic  pairs  of  lenses.     The  eye-piece  usually 
contains  two  plano-convex  lenses,  a  combination  which  is  found 
to  be  the  most  favorable  for  diminishing  the  spherical  aberration, 
and  for  enlarging  the  field  of  view.    For  convenience,  the  direc- 
tion of  the  rays  is,  in  many  instruments,  changed  from  a  vertical 
to  a  horizontal  direction,  by  total  reflection  in  a  right-angled 
prism.    In  Fig.  364,  A  is  the  object ;  B,  C,  and  C',  achromatic 
plano-convex  lenses, 

the    piano  -  concave  FlG- 

part  being  of  flint- 
glass,  the  double-con- 
vex part  of  crown- 
glass,  and  the  two 
parts  fitted  and  ce- 
mented together;  D 
the  right-angled  prism ;  E  the  field-glass,  so  called  because  it 
enlarges  the  field  of  view,  by  bending  the  outer  pencils  so  that 
they  come  within  the  limit  of  the  eye-glass  G\  G  the  eye-glass, 


THE    MAGIC    LANTERN 


419 


converging  the  pencils  to  the  eye  at  H,  while  the  rays  of  each 
pencil  diverge  a  little,  as  from  the  magnified  image  back  of  G. 
The  image  seen  by  the  eye  at  H  fills  the  angle  Iff L. 

713.  Microscopes  for  Projecting  Images.— For  the  pur- 
pose of  forming  magnified  images  on  a  screen,  to  be  viewed  by  an 
audience,  the  microscope  is  modified  in  its  arrangements.     One 
form  for  projecting  transparencies,  whether  paintings  or  photo- 
graphs, is  called  the  magic  lantern.      Another  form,  especially 
adapted  for  the  exhibition  of  small  objects  in  natural  history,  is 
the  solar  microscope* 

Such  instruments  are  valuable  as  means  of  instruction  and 
entertainment,  but  they  are  of  no  use  for  investigation  and  dis- 
covery. 

714.  The  Magic 'Lantern. — It  consists  of  a  box,  represented 
in  Fig.  365,  containing  a  lamp,  and  having  openings  so  arranged 
as  to  permit  the  air  to  pass  freely  through  it,  without  letting  light 
escape.     In  front  of  the  lamp  is  a  tube  containing  a  concentrating 
lens,  (7,  the  painting  on  glass,  B3  and  the  lens,  A,  for  producing 

FIG.  365. 


the  image ;  back  of  the  lamp  may  be  a  concave  mirror  for  reflect- 
ing additional  light  on  the  lens  C.  The  transparency  B  is  a 
painting  on  glass,  and  the  strong  light  which  falls  on  it  proceeds 
through  the  lens  A,  as  from  an  original  object  brilliantly  colored. 
It  is  a  little  further  from  A  than  its  principal  focus,  and  therefore 
the  rays  from  any  point  are  converged  to  the  conjugate  focus  in  a 
real  image,  F,  on  a  distant  screen.  This  image  is  of  course  in- 
verted relatively  to  the  object,  and  therefore,  if  the  picture  B  is 
inverted,  F  will  be  erect.  The  lens  may  be  placed  at  various  dis- 
tances from  B  by  the  adjusting  screw  «,  so  as  to  give  the  greatest 
distinctness  to  the  image  at  any  given  distance  of  the  screen.  Ac- 
cording to  Art.  618,  the  diam.  of  B  :  diam.  of  F : :  A  B  :  A  F;  and 
therefore,  theoretically,  the  image  may  be  as  large  as  we  please. 


420 


LIGHT. 


But  spherical  aberration  will  increase  rapidly  as  the  image  is  en- 
larged, and  even  if  this  evil  could  be  remedied,  the  want  of  ligh  t 
would  render  the  image  too  faint  to  be  well  seen ;  for  the  illumi- 
nation is  as  much  less  than  that  of  the  painting  as  the  area  is 
greater.  Two  magic  lanterns  placed  side  by  side,  may  throw  dif- 
ferent images  on  the  same  ground,  so  as  to  produce  the  effect 
called  dissolving  views. 

715.  The  Solar  Microscope. — This  does  not  differ  in  prin- 
ciple from  the  magic  lantern.  For  illumination  the  solar  or 
electric  light  is  employed,  and  images  are  formed,  not  of  artificial 
paintings,  but  of  small  natural  objects.  The  lens  A  (Fig.  366), 

FIG.  366. 


which  forms  the  image,  is  fixed  in  the  end  of  a  tube,  A  B,  and  at 
the  other  end  is  a  mirror,  M,  which  can  be  turned  on  a  hinge  to 
incline  at  any  angle  with  the  tube.  This  apparatus  is  attached  to 
a  window-shutter,  the  mirror  on  the  outside,  and  the  tube  within. 
By  adjusting  screws  the  mirror  is  inclined  so  as  to  reflect  the  sun- 
beam along  the  tube,  where  it  is  concentrated  by  lenses,  L  L,  upon 
the  object,  0.  Just  beyond  the  object  is  the  lens  A,  of  very  small 
aperture,  by  which  the  image  CD  is  formed.  If  the  sunbeam  is 
large,  and  the  screen  at  a  sufficient  distance,  the  images  of  objects 
may  be  plainly  seen  when  magnified  millions  of  times  in  area. 
Spherical  aberration,  however,  is  considerable ;  and  this  prevents 
the  instrument  from  being  of  service  for  investigation. 

716.  The  Telescope. — The  telescope  aids  in  vieiving  distant 
bodies.  An  image  of  the  distant  body  is  first  formed  in  the  prin- 
cipal focus  of  a  convex  lens  or  a  concave  mirror;  and  then  a 
microscope  is  employed  to  magnify  that  image  as  though  it  were 
a  small  body.  The  image  is  much  more  luminous  than  that 
formed  in  the  eye,  when  looking  at  the  heavenly  body,  because 
there  is  concentrated  in  the  former  the  large  beam  of  light  which 


THE    POWERS    OF    THE    TELESCOPE. 


421 


falls  upon  the  lens  or  mirror,  while  the  latter  is  formed  by  the 
slender  pencil  only  which  enters  the  pupil  of  the  eye.  If  the 
image  in  a  telescope  is  formed  by  a  lens,  the  instrument  is  called 
a  refracting  telescope  ;  but  if  by  a  mirror,  a  reflecting,  telescope. 

717.  The  Astronomical  Telescope. — This  is  the  most 
simple  of  the  refracting  telescopes,  consisting  of  a  lens  to  form  an 
image  of  the  heavenly  body,  and  a  single  microscope  for  magnify- 
ing that  image.  The  former  is  called  the  object-glass,  the  latter 
the  eye-glass.  The  image  is  of  course  at  the  principal  focus  of 
the  object-glass,  and  the  eye-glass  is  placed  at  its  own  focal  dis- 
tance beyond  the  image,  in  order  that  the  rays  of  each  pencil  may 
emerge  parallel ;  therefore  the  two  lenses  are  separated  from  each, 
other  by  the  sum  of  their  focal  distances.  The  lines  marked 
A,  A',  A"  (Fig.  367),  represent  the  cylinder  of  rays  which  flow 


FIG.  367. 


from  the  highest  point  of  the  object,  and  which  cover  the  whole 
object-glass,  M  N.  All  these  rays  are  collected  at  a,  the  lowest 
point  of  the  image,  the  axis  of  the  pencil,  A  a,  being  a  straight 
line  (Art.  615).  After  crossing  afc  «,  they  are  received  on  the 
lower  edge  of  the  eye-glass,  P  Q,  by  which  they  are  made  parallel, 
but  the  entire  pencil  is  bent  toward  the  axis  of  the  lenses,  and 
meets  it  at  F.  The  beam,  £,  B',  B",  coming  from  the  centre  of 
the  object,  forms  the  centre,  #,  of  the  image ;  and  (7,  C",  C",  from 
the  lowest  point  of  the  object,  forms  the  top,  c,  of  the  image.  In 
a  similar  manner  each  point  of  the  image  is  formed  by  the  con- 
centrated rays  which  emanate  from  a  corresponding  point  in  the 
object.  These  innumerable  pencils,  after  diverging  from  their 
focal  points  in  the  image,  are  turned  toward  the  axis  by  passing 
through  the  eye-piece,  while  the  rays  of  each  become  parallel. 
At  F  there  is  a  diaphragm  having  an  aperture,  at  which  the  eye 
is  placed. 

718.  The  Powers  of  the  Telescope.— The  magnifying 
power  of  the  astronomical  telescope  is  expressed  by  the  ratio  of  the 
focal  distance  of  the  object-glass  to  that  of  the  eye-glass.  For  (Fig. 


422  LIGHT. 

367)  the  object,  as  seen  by  the  naked  eye,  fills  the  angle  ADO, 
between  the  axes  of  its  extreme  pencils.  But,  since  the  axes  cross 
each  other  in  straight  lines  at  the  optic  centre  of  the  lens,  ADC 
—  aD c.  Therefore,  to  an  eye  placed  at  the  object-glass,  the 
image,  a  c,  appears  just  as  large  as  the  object ;  while  at  the  eye- 
glass it  appears  as  much  larger  in  diameter  as  the  distance  is  less. 

The  illuminating  power  is  important  for  objects  which  shed  a 
very  feeble  light  on  account  of  their  immense  distance.  This 
power  depends  on  the  size  of  the  beam,  that  is,  on  the  aperture  of 
the  object-glass. 

The  defining  power  is  the  power  of  giving  a  clear  and  sharply 
defined  image,  without  which  both  the  other  powers  are  useless. 
And  it  is  the  power  of  producing  a  well-defined  image  which  lim- 
its both  of  the  other  powers.  For  every  attempt  to  increase  the 
magnifying  power  by  giving  a  large  ratio  to  the  focal  lengths  of 
the  object-glass  and  the  eye-glass,  or  to  increase  the  illuminating 
power  by  enlarging  the  object-glass,  increases  the  difficulties  in 
the  way  of  getting  a  perfect  image.  These  difficulties  are  three — 
the  spherical  aberration  (Art.  621),  the  chromatic  aberration  (Art. 
634),  and  unequal  densities  in  the  glass.  The  third  difficulty  is  a 
very  serious  one,  especially  in  large  lenses.  Very  few  good  object- 
glasses  have  been  made  so  large  as  fifteen  inches  in  diameter. 

719.  Manner  of  Mounting. — The  equatorial  mounting  of 
large  telescopes  is  quite  essential  for  accuracy  of  observation  or 
measurement.    When  the  magnifying  power  is  great,  the  diurnal 
motion  is  very  perceptible,  and  the  body  quickly  leaves  the  field 
of  view.    To  prevent  this,  the  telescope  is  so  mounted  as  to  re- 
volve on  an  axis  parallel  to  the  earth's  axis,  and  then  by  means  of 
a  clock  it  has  a  motion  communicated  to  it,  by  which  it  exactly 
keeps  up  with  the  apparent  motion  of  a  heavenly  body.    Another 
axis,  at  right  angles  with  the  former,  allows  the  telescope  to  be 
directed  to  a  point  at  any  distance  north  or  south  of  the  celestial 
equator. 

Astronomical  telescopes,  when  of  portable  size,  are  usually 
mounted  upon  a  tripod  stand,  and  admit  of  motion  on  a  horizon- 
tal and  a  vertical  axis. 

720.  The  Terrestrial  Telescope. — In  order  to  secure  sim- 
plicity, and  thus  the  highest  excellence,  in  the  astronomical  tele- 
scope, the  image  is  allowed  to  be  inverted,  which  circumstance  is 
of  no  importance  in  viewing  heavenly  bodies.     But,  for  terrestrial 
objects,  it  would  be  a  serious  inconvenience ;  and,  therefore,  a  ter- 
restrial telescope,  or  spy-glass,  has  additional  lenses  for  the  purpose 
of  forming  a  second  image,  inverted,  compared  with  the  first,  and, 
therefore,  erect,  compared  with  the  object.    In  Fig.  368,  m,  m,  m, 


GALILEO'S    TELESCOPE. 

represent  a  pencil  of  rays  from  the  top  of  a  distant  object,  and 
n,  n,  n,  from  the  bottom;  A  B,  the  object-glass;  m'n,  the  first 
image;  CD,  the  first  eye-glass,  which  converges  the  pencils  of 

FIG.  36& 


parallel  rays  to  L.  Instead  of  placing  the  eye  at  L,  the  pencils  are 
allowed  to  cross  and  fall  on  the  second  eye-glass,  E  Fy  by  which 
the  rays  of  each  pencil  are  converged  to  a  point  in  the  second 
image,  m'  n',  which  is  viewed  by  the  third  eye-glass,  G  H.  The 
second  and  third  lenses  are  commonly  of  equal  focal  length,  and 
add  nothing  to  the  magnifying  power. 

Such  instruments  are  usually  of  a  portable  size,  and  hence  the 
aberrations  are  corrected  with  comparative  ease,  by  the  methods 
already  described.  The  spy-glass,  for  convenient  transportation, 
is  made  of  a  series  of  tubes,  which  slide  together  in  a  very  com- 
pact form. 

721.  Galileo's  Telescope.— This  was  the  first  form  of  tele- 
scope, having  been  invented  by  Galileo,  whose  name  it  therefore 
bears.  It  differs  from  the  common  astronomical  telescope  in 
having  for  the  eye-glass  a  concave  instead  of  &  convex  lens,  which 
receives  the  rays  at  such  a  distance  from  the  focus  to  which  they 
tend,  as  to  render  them  parallel.  Thus,  the  rays,  M,  M,  M  (Fig. 
369),  from  the  top  of  the  object,  are  converged  by  the  object-glass, 

FIG.  369. 


A  B,  toward  m,  in  the  image ;  and  the  pencil,  N,  N,  N,  from  the 
bottom  of  the  object,  is  converged  toward  n ;  but  the  concave  lens 
CD  is  interposed  at  such  a  point  as  to  render  these  converging 
rays  parallel,  and  in  this  way  they  come  to  the  eye  situated  behind 
the  lens.  But,  though  the  rays  converge  before  they  reach  the 
concave  lens,  the  pencils  diverge,  having  crossed  at  F\  therefore, 


424  LIGHT. 

in  passing  the  concave  lens,  they  are  made  to  diverge  more,  and 
will  enter  the  eye  as  if  they  had  crossed  at  a  much  nearer  point 
than  F.  The  angle  between  these  extreme  pencils  is  the  angle 
which  the  object  appears  to  fill ;  and  the  magnifying  power  is  in 
the  ratio  of  this  angle  to  the  angle  MFN—mFn\  and  that 
equals  the  ratio  of  the  focal  distance  of  A  B  to  the  focal  distance 
of  CD.  The  object  appears  erect  in  the  Galilean  telescope,  since 
the  pencil,  which  comes  from  the  top  of  the  object,  appears  to 
come  from  the  top  of  the  virtual  image ;  thus,  the  parts  of  the 
object  and  image  are  similarly  situated.  It  is  obvious  that,  since 
the  pencils  diverge,  only  the  central  ones,  within  the  size  of  the 
pupil,  can  enter  the  eye.  This  circumstance  exceedingly  limits 
the  field  of  view,  and  unfits  the  instrument  for  telescopic  use.  It 
is  employed  for  opera-glasses,  having  a  power  usually  of  only  two 
or  three  in  diameter. 

722.  The  Gregorian  Telescope.— This  is  the  most  frequent 
form  of  reflecting  telescope,  and  receives  its  name  from  the  inven- 
tor, Dr.  Gregory,  of  Scotland.  The  light  from  a  heavenly  body, 
entering  the  open  tube  (Fig.  370),  is  received  on  the  large  concave 

FIG.  370. 


speculum,  E>  which  forms  an  inverted  image,  m,  at  the  principal 
focus ;  the  rays  of  each  pencil  crossing  there  next  meet  the  small 
concave  mirror  F>  which  forms  an  erect  image,  n,  at  the  conjugate 
focus,  beyond  the  speculum,  the  centre  of  the  latter  being  perfo- 
rated to  let  the  light  pass  through.  The  eye-glass,  G,  magnifies 
this  image.  To  avoid  confusion,  only  two  rays  are  drawn  in  the 
figure,  and  those  belong  to  the  central  pencil.  Eays  from  the  top 
of  the  object  would  enter  the  tube  inclining  slightly  downward, 
and  be  reflected  to  the  bottom  of  m,  and  again  to  top  of  n.  Eays 
from  the  bottom  would  ascend,  and  be  reflected  to  the  top  of  the 
first  image,  and  to  the  bottom  of  the  second. 

723.  The  Herschelian  Telescope.— Sir  William  Herschel 
modified  the  Gregorian  by  dispensing  with  the  small  reflector  F9 
and  inclining  the  large  speculum  E,  so  as  to  form  the  image  near 
the  edge  of  the  tube,  where  the  eye-glass  is  attached.  Thus,  the 
observer  is  situated  with  his  back  to  the  object.  The  speculum  of 
Herschel's  telescope  was  about  four  feet  in  diameter,  and  weighed 
more  than  2,000  pounds,  and  its  focal  length  was  forty  feet.  The 
Earl  of  Eosse  has  since  constructed  a  Herschelian  telescope  having 
an  aperture  of  six  feet,  and  a  focal  length  of  fifty  feet. 


APPENDIX. 


APPLICATIONS  OF  THE  CALCULUS. 

I.  FALL  OF  BODIES. 

1.  Differential  Equations  for  Force  and  Motion.—  These 

are  three  in  number,  as  follows  : 


ds 

v  = 

<•> 


v  =  -j-f 
d  t 


~~  d  t  ~  d  f 
3.  fds  =  v  civ. 

These  equations  are  readily  derived  from  the  elementary  prin- 

o 

ciples  of  mechanics.    In  Art.  6  we  have  v  =  -.     Eeducing  the 

t 

numerator  and  denominator  to  infinitesimals,  v  remains  finite,  and 

ds 

the  equation  becomes  v  =  -~-\  which  is  Equation  1st.    Therefore, 

if  the  space  described  by  a  body  is  regarded  as  a  function  of  the 
time,  the  first  differential  coefficient  expresses  the  velocity. 

v 

Again  (Art.  12),  /  =  -,  where  /  represents  a  constant  force. 
t 

Making  velocity  and  time  infinitely  small,  we  get  the  intensity  of 
the  momentary  force,  /=  -j->     But,  by  Equation  1st,  v  =  -j-.  ; 


.'.f=  -,-75;  which  is  Equation  3d.    Hence  we  learn  that  the  first 
cl>  t 

differential  coefficient  of  the  velocity  as  a  function  of  the  time,  or 
the  second  differential  coefficient  of  the  space  as  a  function  of  the 
time,  expresses  the  force. 

Equation  3d  is  obtained  by  multiplying  the  1st  and  2d  cross- 
wise, and  removing  the  common  denominator. 

We  proceed  to  apply  these  equations  to  the  preparation  of  for- 
mulae for  falling  bodies. 

2.  Bodies  falling  through  Small  Distances  near  the 
Earth's  Surface.—  In  this  case,  let  the  accelerating  force,  which 


426 


APPENDIX. 


4) 

•is  considered  constant,  be  called  g.  Then,  by  Eq.  2,  g  =  — , .-.  dv 
=  g  d  t.  Integrating,  we  have  v  =  g  t  +  C.  But,  since  v  =  0 
when  t  =  0,  /.  v  —  g  t,  and  t  =  -,  as  in  formulas  5,  6,  Art.  28. 

c/ 

Again,  substituting  g  t  for  v  in  Eq.  1,  ds  =  gtdt;  and  by 
integration,  5  =  \g  f  +C;  but  (7=0,  for  the  same  reason  as  be- 


fore;  .:  s  =  ±gt*,  and  #  =  y  —  ,  as  in  formulas  1,  2,  Art.  28. 

c/ 

Once  more,  equating  .the  two  foregoing  values  of  t,  we  have 

_  v* 

v  —  v%  gs,  and  s  =  ^-,  as  in  formulas  3,  4,  Art.  28. 


If,  in  the  equation,  s  =  £  0r  f  ,  v  be  substituted  for  #  £,  we  have 
s  =  -|-i;^  or#£  =  2s;  that  is,  the  acquired  velocity  multiplied  by 
the  time  of  fall  gives  a  space  twice  as  great  as  that  fallen  through 
(Art.  25). 


FIG.  1. 


3.  Bodies  falling  through  Great  Distances, 
so  that  Gravity  is  Variable,  according  to  the 
Law  in  Art.  16.  — 

Suppose  a  body  to  fall  from  A  to  B  (Fig.  1),  to- 
ward the  centre  (7.  Let  AG—a\  B  G—x\  D  C—r, 
the  radius  of  the  earth. 

The  force/  at  B,  is  found  by  the  principle,  Art.  16, 


4.  To  find  the  Acquired  Velocity  .—Substitute  g  r*  x~*  for 
/,  and  a  —  x  for  s,  in  Equation  3d,  and  we  have  g  r*  x~*  .d(a  —  x) 
=  v  d  v ;  /.by  integration  j-  v*  =  f  —  g  r*  x~*  d  x  =  g  r*  x~l  +  C. 
But  v  =  0,  when  x  —  a ;  /.  C  •=  —  g  r*a~l ;  and 


ax 


ax 


This  is  the  general  formula  for  the  acquired  velocity.    If  the 
body  falls  to  the  earth,  x  —  r,  and  the  formula  becomes 


FALL    OF    BODIES. 


427 


Again,  if  the  body  falls  to  the  earth  through  so  small  a  space 
that  -  may  be  regarded  as  a  unit,  the  formula  reduces  to 


the  same  as  obtained  by  other  methods. 

If  a  body  falls  to  the  earth  -from  an  infinite  distance,  it  does 
not  acquire  an  infinite  velocity.    For  then,  as  we  may  put  a  for 


(2  .  32|  .  3956  .  5280)*  feet  =  6.95  miles. 

Therefore,  the  greatest  possible  velocity  acquired  in  falling  to 
the  earth  is  less  than  seven  miles  ;  and  a  body  projected  upward 
with  that  velocity  would  never  return. 

5.  To  find  the  Time  of  Falling.  —  From  equation  first  we 

,,  .     7,    ds  .  ,   2tfrs#—  # 

obtain  a  t=  —  ;  in  this,  substitute  d(a—x)  for  as,  and 

for  v,  as  found  in  the  preceding  article  ;  then 

—  x)  __  /    a   \l   —  x$ 
^r*'   " 


(ax) 


-*' 


(a  -a?) 


{2  g  r>  (a  -  x)}* 

.-.  by  integration  t  -  (o^p)~  •  J  -  ^dx(a-  3)-*. 

j 

By  the  formula  in  the  calculus  for  reducing  the  index  of  x  we 
obtain 


a-  a?)-    =  (ax-  x*)    -    vers~l   —    +  C. 


Now,  when  t  =  0,  x  —  «;  .*.  (7  =  -^  : 

<0 


6.  Bodies  falling  within  the  Earth  (sup- 
posed to  be  of  uniform  density),  where 
Gravity  Varies  as  the  Distance  from  the 
Centre.  — 

Suppose  a  body  to  fall  from  A  to  B  (Fig.  2)  ; 
and  let  D  C  =  r,  A  C  —  a,  and  B  C  =  x.  Then 

r:x::g:f=-x  =  force  at  B. 


428  APPENDIX. 

To  find  the  velocity  acquired.  —  By  Eq.  3d, 

v  dv  =fds;  .*.  vd  v  —  -x.d  (a  —  x)  =  —  ^—   -; 

a  x* 
/.  £  v*=  —  ~—  -f  C\  but  v  =  0  when  x  —  a; 


'If  the  body  falls  from  the  surface  to  the  centre,  x  —  0,  and 

this  formula  becomes  v  =  (gr)^  =  (32  ;J  x  3956  x  5280)^  =  25,904= 
feet  per  second. 

To  find  the  time  of  falling.—  -By  Equation  1st,  and  substitu- 

,.  ,,  .     7,      ds      d(a—x)  dx  —dx 

tions,  we  obtain  d  t  =  —  =  —  -  --  -  =  --  =  —  -,  -- 


When  t  =  0,  #  =  «,  -  =  1,  and  the  arc,  whose  cosine  is  1  =  0; 
...  (7=0.    .'.t= 


If  the  body  falls  to  the  centre,  x  =  0,  and  £=(-)    x-;in 

v//         A 

which  a  does  not  appear  at  all  ;  so  that  the  time  of  falling  to  the 
centre  from  any  point  within  the  surface  is  the  same  ;  and  equals 


3956  x 

--  ^n  ---  )    x  1.570796  in  seconds,  or  21m.  5.8s. 


'«J 


IL  CENTRE  OF  GRAVITY. 

7.  Principle  of  Moments.— In  order  to  apply  the  processes 
of  the  calculus  to  the  determination  of  the  centre  of  gravity,  the 
principle  is  used,  which  was  proved  (Art.  78),  that  if  every  par- 
ticle of  a  body  be  multiplied  by  its  distance  from  a  plane,  and 
the  sum  of  the  products  be  divided  by  the  sum  of  the  particles, 
the  quotient  is  the  distance  of  the  common  centre  from  the  same 
plane.    The  product  of  any  particle  or  body  by  its  distance  from 
the  plane,  is  called  its  moment  with  respect  to  that  plane. 

8.  General  Formulee. — Let  BAG  (Fig.  3)  be  any  symmetri- 
cal curve,  having  A  X  for  its  axis  of  abscissas,  and  A  Y,  at  right 


APPLICATION    OF    FORMULAE. 


429 


FIG.  3. 


angles  to  it,  for  its  axis  of  ordinates.  It  is  obvious  that  the 
centre  of  gravity  of  the  line  BAG,  of  the  area  B  A  C,  of 
the  solid  of  revolution  around  the  axis 
A  X,  and  of  the  surface  of  the  same 
solid,  are  all  situated  on  A  X,  on  ac- 
count of  the  symmetry  of  the  figure. 
It  is  proposed  to  find  the  formula  for 
the  distance  of  the  centre  from  A  Y, 
in  each  of  these  cases.  Let  G  in  every 
instance  represent  the*  distance  of  the 
genera]  centre  of  gravity  from  the  axis 

A  Y,  or  the  plane  A  Y,  at  right  angles  to  A  X.  The  distance  G 
would  plainly  be  the  same  for  the  half  figure  B  A  D,  as  for  the 
Whole  SAC;  expressions  may  therefore  be  obtained  for  either, 
according  to  convenience. 

1.  The  line  A  B. — Let  x  be  the  abscissa,  and  y  the  ordinate ; 

then  (dx>  +  dy*Y  is  ttie  differential  of  the  line  A  B.  For  brevity, 
let  s  —  the  line,  and  d  s  its  differential.  If  we  now  multiply  this 
differential  by  its  distance  from  A  Y,  x  d  s  is  the  moment  of  a 
minute  portion  of  the  line ;  and  the  integral  of  it,  f  x  d  s,  is  the 
moment  of  the  whole.  Dividing  this  by  the  line  itself,  i.  e.  by  s, 


we  have 


fxds 


for  the  distance  G. 


2.  The  area  B  A  D. — The  differential  of  the  area  is  y  d  x ;  the 
differential  of  its  moment  is  x  y  d  x ;   hence  the  moment  itself  is 

/  x  y  d  x :  and  the  distance  G  =  - — ~ — . 

area 

3.  TJie  solid  of  revolution. — The  differential  of  the  solid,  gen- 
erated by  the  revolution  of  A  B  on  A  X,  is  n  y*d  x ;  the  differen- 
tial of  its  moment  is  TT  x  y*d  x ;  and  the  moment  is  /  TT  x  y*d  x ; 

hence  the  distance  G  =  - — —^ . 

solid 

4.  The  surface  of  revolution. — The  differential  of  the  surface  is 
2  TT  y  d  s ;  the  differential  of  its  moment  is  2  TT  x  y  d  s ;  and  there- 
fore the  moment  isf^nxi/ds:  and  the  distance  G  —  -~ — . 

J  surface 


9.  Application  of  Formulae.  —  "We  proceed  to  determine 
the  centre  of  gravity  in  a  few  cases  by  the  aid  of  these  formulae  : 

1.  A  straight  line.  —  Imagine  the  line  placed  on  A  X,  with  one 
extremity  at  the  origin  A.  The  moment  of  a  minute  part  of  it  is 
zdx,  and  that  of  the  whole  is  /  x  d  x,  while  the  length  of  the 


1    i    •  ^ 

whole  is  x  ;  .*.  G  = 


fxdx      \  x1  +  0 


..     .  -,     ,,     -,      ,T 
=  £  x,  as  it  evidently  should 


430  LIGHT. 

be.    In  all  the  cases  considered  here,  C—  0,  because  the  function 
vanishes  when  x  does. 


2.  The  arc  of  a  circle.—  By  formula  1st  we  have#  =    --  but 

s 

d  s  —  (d  x*+  dtf)1*  \  by  the  equation  of  the  circle,  if—  2  a  x  —  x*  ; 

,         ,          N   ,          ,   3     (a  —  xfdx*  * 

.:  ydy  =  <«-*)  dx;  ,.  df=  *—    — 


adx  a  r       xdx 

— -  =-  l  vers 


—(2ax  —  x*Y  !•=-(*  —y)  =  a  —~  =  «  —  ^r,  if  the  arc  is  dou- 

)  5  S  t 

bled  and  called  t,  and  c  (chord)  put  for  2  y.    As  a  —  ~  is  the  dis- 

t 

tance  from  the  origin  A9  and  a  =  radius  of  the  arc  ;  .*.  the  distance 
from  the  centre  of  the  circle  to  the  centre  of  gravity  of  the  arc, 

fi  (* 

is  —  ,  which  is  a  fourth  proportional  to  the  arc,  the  chord,  and  the 

radius. 

When  the  arc  is  a  semi-circumference,  c  ==  2  a,  and  t  =  TT  a  ; 
/.  the  distance  of  the  centre  of  gravity  of  a  semi-circumference 

from  the  centre  of  the  circle  is  —  . 

7T 

3.  The  area  of  a  circular  sector.  —  Suppose  the  given  sector  to 
be  divided  into  an  infinite  number  of  sectors  ;  then  each  may  be 
considered  a  triangle,  and  its  centre  of  gravity  therefore  distant 

from  the  centre  of  the  circle  by  the  line  -=-.  Hence  the  centres  of 

o 

gravity  of  all  the  sectors  lie  in  a  circular  arc,  whose  radius  is  -^-  ; 

o 

so  that  the  centre  of  gravity  of  the  whole  sector  coincides  with 
the  centre  of  gravity  of  that  arc.  The  distance  of  the  centre  of 
gravity  of  the  arc  from  the  centre  of  the  circle,  by  the  preceding 

.2         2         2,      2ac     ,.,.,, 

case,  is^a  x  -c-^-£  =  -^-r-,  which  is  therefore  the  distance  of 
o          o          o  6  1 

the  centre  of  gravity  of  the  sector  from  the  centre  of  the  circle. 

When  the  sector  is  a  semicircle  the  distance  becomes  —  ^  --- 

&  ~  a 


I 


CENTRE    OF    GRAVITY.  431 

4.  Tfie  area  of  a  parabola.  —  The  equation  of  the  curve  is 

y*=px,OYy=p*x*; 
therefore  the  formula  2  for  moment, 


but  the  area  of  the  half  parabola  =  %p*x%. 

.-.  G  =  |  p*  &  -f-  \$  x%  =  |  x. 

To  find  the  distance  of  the  centre  of  gravity  of  the  semi-parab- 
ola from  the  axis  A  X,  proceed  as  follows  :  The  differential  of  the 
area,  as  before,  equals  y  d  x  ;  and  the  distance  of  its  centre  from 
A  X  is  £  y.  Therefore  its  moment  with  respect  to  A  X  is  \  y*  d  x 
=  £  p  x  d  x  ;  and  the  moment  of  the  whole  is  f^p  x  d  x  =  |  p  x9  ; 
/.  the  distance  of  the  centre  from 

A  X—  \p  x9  -f-  lp^x%  —  |/2-  #2  =  |  yt 
5.  The  area  of  a  circular  segment.  —  The  equation  of  the  circle 
is,  y  =  (2  a  x  —  x*)?.    Therefore  (formula  2), 

fxydx=fx(2ax-  a?)*  d  x. 

Add  and  subtract  a  (2  a  x  —  x*)*  d  x,  and  it  becomes 

fa  (2  ax  -  x^dx  -f(a  -  x)  (2ax  — 

(2  ax—  a*)*  (a—  x)  dx 
^ 


.  (2  ax-*)* 


"When  x  =  a,  G  =  a  —  ^-}  and  the  distance  of  the  centre  of 

O  7T 

gravity  of  a  semicircle  from  the  centre  of  the  circle  =  ~.    When 

O  7T 

x  =  2a,  G  =  a,  as  it  plainly  should  be. 

6.  A  spherical  segment.  —  The  equation  of  the  circle  is  y2  = 
2  a  x  —  x*.    Therefore  (formula  3), 


"When  x  =a,  G  —  f  a  ;  that  is,  the  centre  of  gravity  of  a  hem- 
isphere is  |  of  radius  from  the  surface,  or  |  of  radius  from  the 
centre  of  the  sphere.  If  x  =  2  a,  G  =  a. 

7.  A  right  cone.  —  In  this  case  A  B  (Fig.  3),  is  a  straight  line, 
and  its  equation  is  y  —  a  x,  where  a  is  any  constant. 


432 


APPENDIX. 


=  fira'x'dx  =  TaV  ;  /.  £ 

•  4t 

Hence  the  centre  of  gravity  of  a  cone  is  three-fourths  of  the  axis 
from  the  vertex.     See  Art.  75. 

8.  The  convex  surface  of  a  right  cone.  —  The  equation  is 

y  =  ax;  .:  dtf  =  tfdx*;  and  (dx*  +  dy^  =  (a?  +  l)?dx. 
Therefore  (formula  4), 


=  the  moment  of  the  surface.     The  surface  itself, 
=  *  y  (tf  +  y^  =  nax*(a*  +  1)*.     ,.  ^  » 


The  centre  of  gravity  of  the  convex  surface  of  a  right  cone  is  on 
the  axis,  at  a  distance  equal  to  two-thirds  of  its  length  from  the 
vertex. 

III.  CENTEE  OF  OSCILLATION. 

9.  To  find  the  Moment  of  Inertia  of  a  Body  for  any 
given  Axis. — To  render  the  formula  I  =     ^r  -,   •  suitable  to  the 


application  of  the  calculus,  we  have  simply  to  substitute  the  sign 
of  integration  for  8,  and  d  M  for  m,  and  we  have 


Mk 


(i) 


It  is  useful  to  know  how  to  find  the  moment  of  inertia  with  respect 

to  any  axis  by  means  of  the  FIG.  4. 

known  moment  with  respect  to 

another  axis  parallel  to  it  and 

passing  through  the  centre  of 

gravity  of  the  body. 

Let  A  Z  (Fig.  4)  be  the  axis 
passing  through  the  centre  of 
gravity  of  the  body  for  which 
the  moment  of  inertia  is  fr*dM, 
and  let  A'  Z'  be  the  axis  paral- 
lel to  it,  for  which  the  moment 
of  inertia,  ,/V 2  d  M  of  the  same 
mass  M9  is  to  be  determined. 
For  every  particle  m  of  the  body 
the  corresponding  value  of  A  m' 
is  r*  =  x*  +  y3.  In  like  man- 


CENTRE    OF    OSCILLATION.  433 

ner,  if  we  denote  the  co-ordinates  of  A'  by  a  and  j3,  and  the  dis- 
tance between  the  axes  by  a,  we  shall  have  #3  =  aa  +  /32.    Now  the 
distance  of  the  particle  m  from  A'  Z'  is  rn  =  (a?  —  a)a+  (^  —  (3)* 
=  x*  +  y*  +  a?  +  (3*  —  2ax-2l3y  =  r''  +  a*  —  2ax  —  2(3y,.\ 
frn  d  M=  fr*dM  +  a*fd  H-  2afxdM-2ftfydM  =  a'  M 
+  fr*dM,.  ......        .        .        (2) 

since  A  Z  passes  through  the  centre  of  gravity  of  the  body.  Hence, 
the  moment  of  inertia  of  a  body  with  respect  to  any  axis  is  equal  to 
the  moment  of  inertia  with  respect  to  a  parallel  axis  through  the 
centre  of  'gravity  r,  plus  the  mass  of  the  body  multiplied  by  the  square 
of  the  distance  between  the  two  axes. 

Put  C  —  the  moment  of  inertia  with  respect  to  an  axis  through 
the  centre  of  gravity  ;  then  the  distance  from  the  axis  of  suspen- 
sion to  the  centre  of  oscillation,  the  axes  being  parallel,  will  be 


10.  Examples.  — 

1.  Find  the  centre  of  oscillation  of  a  slender  rod  or  straight 
line  suspended  at  any  point. 

Let  a  and  b  be  the  lengths  on  opposite  sides  of  the  axis  of  sus- 
pension, then  by  (1) 

_  fr*dM  _          fr*dr  2  (a3  .+  b9)      2(a't-ab  +  V) 

Mk      ~(a  +  b)±(a-b)~3(a'-b'l)~        3  (a  -  b) 

between  the  limits  r  =  +  a  and  r  —  —  b. 

If  the  rod  is  suspended  at  its  extremity,  5  =  0,  and  l  =  \a.    If 
it  is  suspended  at  its  middle  point,  a  =  b  and  I  —  oc  . 

2.  Find  the  centre  of  oscillation  of  an  isosceles  triangle  vibra- 
ting about  an  axis  in  its  own  plane  passing  through  its  vertex. 

Put  b  and  h  for  the  base  and  altitude  of  the  triangle;  then  by 


If  the  axis  of  suspension  coincides  with  the  base  of  the  trian- 


g]e,thenZ= 


3.  Find  the  centre  of  oscillation  of  a  circle  vibrating  about  an 
axis  in  its  own  plane. 

C  =fr*  dM=  2fx*  ydx  =  2fx>  (  JP  - 


28 


434 


APPENDIX. 


Taking  this  integral  between  x  =  —  r  and  x  =  -f  r,  we  have 
r  -  —  ^^  _  !L?* 

=   2  '    2  4   * 

Substituting  this  value  of  (7  in  (3)  we  have 
TT  R* 


_. 


4.  Find  the  centre  of  oscillation  of  a  cir- 
cle vibrating  about  an  axis  perpendicular 
to  it. 

Let  K  L  (Fig.  5)  be  an  elementary  ring 
whose  radius  is  x  and  whose  breadth  is  d  x  ; 
then 


.  2  nx  dx 


dM  =  2nxdz,  and  C  = 

^+037r.#' 


r>a  r>a 

As  a  +  ^ L  is  greater  than  a  +  j— ,  a  cir- 
£>  a  4  ct> 

cular  pendulum  will  vibrate  faster  when  the 
axis  of  suspension  is  in  its  plane,  than  when 
it  is  perpendicular  to  it. 


FIG.  5. 


IY.  CEOTBE  OF  HYDKOSTATIC  PKESSUKE. 

11.  General  Formula. — Let  the  surface  pressed  upon  be 
plane  and  vertical ;  and  let  the  water  level  be  the  plane  of  refer- 
ence. Suppose  the  surface  to  have  a 
symmetrical  form  with  reference  to  a  Fick6. 

vertical  axis,  x,  whose  ordinate  is  y 
(Fig.  6).  A  horizontal  element  of  the 
surface  is  2  y  d  x,  and  (since  the  pres- 
sure varies  as  the  depth)  the  pressure 
on  that  element  2  x  y  d  x.  Hence  the 
whole  pressure  to  the  depth  x  is 
S%xydx=%fxydx.  The  mo- 
ment of  the  pressure  on  the  element 

of  surface  is  2  x*  y  d  x\  and  the  sum  of  all  the  moments  to  the 
same  depth  is  /  2  a3  y  dx  —  2  f  tf  y  d  x.    Therefore,  putting  p 

for  the  depth  of  the  centre  of  pressure,  p  =  ~-^~. 

f  xy  dx 


CENTRE    OF    HYDROSTATIC    PRESSURE.        435 

12.  Examples. 

1.  A  rectangle.  —  Let  its  height  =  Ji,  and  its  base  =  b  ;  then  2  y 
everywhere  equals  b,  and  a  horizontal  element  at  the  depth  x  is 
b  d  x,  the  pressure  on  it  is   ~b  x  d  x,  and  the  moment  of  that 
pressure  is  b  x*  d  x  ;  /.  the  depth  of  the  centre  of  pressure  p  = 

fbx*dx      \bx*  +  c 

^rj-  —  -j—  =  7  .    a  ,  —  •    Since  the  pressure  and  area  is  each  zero, 

/  bxdx       -±  bx*  +  c 

when  x  is  zero,  c  and  c'  both  disappear,  and  p  =  f  x,  which  for  the 
whole  surface  becomes  p  —  f  h.  That  is,  the  centre  of  pressure  on 
a  vertical  rectangular  surface  reaching  to  the  water  level,  is  two- 
thirds  of  the  distance  from  the  middle  of  the  upper  side  to  the 
middle  of  the  lower. 

2.  A  triangle  whose  vertex  is  at  the  surface  of  the  water,  and  its 
base  horizontal.  —  Let  the  triangle  be  isosceles,  its  height  =  h,  and 


J 
its  base  =  l\  then  h:bi:  x:%  y  =  •=•  x.   Therefore^  = 

j  £« 
=  -  —  3  =  |  x  ;  and  for  the  whole  height,  |  h. 

3  X 

If  the  triangle  is  not  isosceles,  it  may  be  easily  shown  that  the 
centre  of  pressure  is  on  the  line  joining  the  vertex  and  the  middle 
of  the  base,  at  a  distance  from  the  vertex  equal  to  three-fourths  of 
the  length  of  that  line. 

3.  A  triangle  whose  base  is  at  the  water  level  —  Then  h  :  b 

::  h  —x:%  y  =  b  —  ^  x.  Therefore  the  pressure  is  /—  bxdx 
—  /  -—  j-  x*  d  x,  because  d  x  is  negative.  The  moment  of  the 
pressure  is  /—  b  x*  d  x  —  /—  -=  x*dx. 


x3  dx      -tf  -f 
Therefore  p  = 


-fbx  dx  +  J  ^ 
—  3  z4 


3  ;  and,  when  x  =  h,  this  becomes  -A  h. 

b  Ji  x  —  &  x          b  /i  —  4:X 

In  general,  the  centre  of  pressure  is  at  the  middle  of  the  line  join- 
ing the  vertex  and  the  middle  of  the  base. 

4.  A  parabola  whose  vertex  is  at  the  surface.  —  As  y  =  p'2  x*, 

fx*p?xldx      fx'dx      §x$      5        .5,, 
therefore  p  =  —   •*-  —  j  -  =  --  -  -  =    —  =  ^x;  or  =  h,  for 

fx^x^dx      fx*dx      \x*> 
the  whole  area. 


436  APPENDIX. 

5.  A  parabola  whose  lase  is  at  the  surface.  —  As  7i  —  x  is  the 

i     ,T     »        i  i.  —  /  (h  —  #)2z2  ^  # 

depth  of  an  element,  d  x  is  negative,    p  =  -  -  = 

-f(h-x)  x^dx 
f  (h9  x?  d  x  -  2  h  x*  d  x  +  a?  d  a?)  _  jA'aJ  -  |  &  s%  +  f  x*  _ 


2  —  1  ^  a?  +  4  a" 
r-s-         -2  —  ;  and  when  x  —  ^  the  expression  becomes 


-  j 


+  4  A2  _  4 


V".  ANGULAR  RADIUS  OF  THE  PRIMARY  AND  SECONDARY  EAIN- 

BOW  AND  THE  HALO. 

13.  The  Primary  Rainbow. — Since  the  primary  bow  is 
formed  by  those  rays  which,  on  emerging  after  one  reflection, 
make  the  largest  angle  with  the  incident  rays,  proceed  to  find 
what  angle  of  incidence  will  cause  the  largest  deviation  of  the 
emerging  rays. 

In  Fig.  7,  let  x  =  angle  of  inci-  FIG.  7. 

dence ;  y  =  angle  of  refraction ;  z  = 
angle  of  deviation ;  n  —  index  of  re- 
fraction. Then,  in  the  quadrilateral 
BDGK,  DBK=DGK=x-y, 
angle  at  D  =  360  -  2  y ;  .-.  K  —  z  — 


.         _ 

•  .    -^  —   —  —  j  -    —  /C  —  v» 

dx       dx 
But  sin  x  =  n  sin  y\ 

7  j         *  &y        cos  x 

/.  cos  x  d  x  =  n  cos  y  d  y,  and  —  =  —    —  . 

dx      ncosy 

.   ...   ,.       4  cos  x 
By  substitution,  -       -  =  2. 
ncos  y 

.*.  2  cos  x  =  n  cos  ^;  and  4  cos*  x  =  ri*  cos2 
But  sin2  #  =  n2  sin2  y  ; 

.\  3  cos2  x  +  1  =  ri*  ;  since  sin2  -H  cos2  —  1. 


.-.  cos  z  •= 


If  1.33  and  1.55,  the  values  of  n  for  extreme  red  and  violet,  be 
used  in  this  formula,  we  obtain  x,  and  therefore  y  and  z,  for  the 
limiting  angles  of  the  primary  bow. 


KADIUS    OP    RAINBOW    AND    HALO. 


437 


14.  The  Secondary  Bow.— To  find  the  angle  of  minimum 
deviation.    Using  the  same  notation  as  before,  we  have  in  the 
pentagon  G  EDBK  (Fig.  8),  G  =  B  — 
180  —  x  +  y\  fi=D  =  2y;  .'.  K=z= 
180  +  2x  —  6; 


FIG.  8. 

G 


"  dx 
6  cos  x 


dx 


=  2 ;  and  3  cos  x  =  n  cos  y ; 
ncos  y 

.-.  9  cos2  x  =  ri*  cos2  y ; 
but     sin8  x  =  ri1  sin2  y ; 

/.  8  cos2  x  +  1  =  n* ; 

.*.    COS  X  — 

which,  as  before,  will  furnish  z  for  each  limiting  color  of  the  sec- 
ondary bow. 

15.  The  Common  Halo.— Let  D  E  (Fig.  9)  be  the  ray  from 
the  sun,  and  F G  the  emergent  ray.     Let  D Ep  =  x\  KEF  —  y\ 


z  =  x  -  y  +  y1  -  x'.     Now,  FI»-  »• 

y  +  x'  =pr  KF-  0=6Q°. 

.•.Z  =  X+y'-C. 

sin  x  —  n  sin  y, 
and  sin  y'—  n  sin  x' ; 

/.  x  =  sin"1  (n  sin  y\ 
and       y'  =  sin"1  (n  sin  #')  = 

sin"1  \n  sin  (7—  ^K 
By  substitution, 

z  —  sin"1  (n  sin  #)  +  sin"1  { n  sin  ( (7  —  y ) }  —  (7.    Therefore  2  is  a 
function  of  y ;  and,  by  differentiating,  we  have 
d  z  _        n  cos  y  n  cos  ( G  —  y) 

dy~  Vl  —  tf'sw*  y       Vl  —  w2sin2  (C—y)~ 
n*  cos2  y  n9  cos3  ( C  —  y ) 

.  1  - sina  y 


.''  1  -  rc8n2  y  "    1  -  ^  sin2  (C-y)y 
.-.  (ri>  -  1)  sin2  y  =  (w1  -  1)  sin2  (C7 -  y); 
.-.  y  =  (7  -  y,  and  y  =  }  Cf; 

and  a/  =  £  (7. 

Hence,  the  minimum  deviation  occurs  when  the  ray  within  the 
crystal  is  equally  inclined  to  the  sides.  Knowing  w,  the  index  of 
refraction  for  ice,  x,  and  its  errial,  /,  can  be  obtained,  and  then  z, 
the  deviation  required. 


// 


9S2906 


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